Second Seminar on Algebra and its Applications · Second Seminar on Algebra and its Applications...

Second Seminar on

Algebra and its ApplicationsArdabil, August 30-September 1, 2012

Organized by the

University of Mohaghegh Ardabili

Organizing Committee

M. Ganji, Chancellor of the UniversityJ. Razmjou, Vice-Chancellor of Research and Technology of the UniversityA. Borhanifar, Vice-Chancellor of Student AffairsGh. Ghasemi, Advisor to the Chacellor of the UniversityY. Abbaspour, Director of Research Affairs of the UniversityS. Parsi, Dean of the Faculty of Mathematical SciencesK. Haghnejad, Deputy Dean of the Faculty of Mathematical SciencesM. Zarebnia, Deputy Dean of the Faculty of Mathematical SciencesH. Abdolzadeh, University of Mohaghegh ArdabiliJ. Azami, University of Mohaghegh ArdabiliA. P. Kazemi, Chairman of the Scientific CommitteeA. Khojali, University of Mohaghegh ArdabiliA. Yousefian Darani, Chairman of the Organizing Committee

Scientific Committee

H. Abdolzadeh, University of Mohaghegh ArdabiliJ. Azami, University of Mohaghegh ArdabiliA. P. Kazemi, Chairman of the Scientific CommitteeA. Khojali, University of Mohaghegh ArdabiliA. Yousefian Darani, Chairman of the Organizing Committee

Invited Speakers

Ayman Badawi, University of Sharjah, United Arab EmiratesEbadollah S. Mahmoodian, Sharif University of Technology, IranReza Naghipour, University of Tabriz, IranPeyman Nasehpour, Osnabrueck University, GermanyAbbas Nasrollah Nejad, Institute for Advanced Studies in Basic SciencesSiamak Yassemi, University of Tehran, Iran

WELCOME TO ARDABIL’2012

We thank all of you for coming and wish you a pleasant and mathematically

successful meeting in Ardabil. The Second Seminar on Algebra and its Applications

is organized by the University of Mohaghegh Ardabili, hosted by the Faculty of

Mathematical Sciences. This seminar has come a short way from its first meeting,

Workshop on Algebra and its Applications, in the University of Mohaghegh Ardabili

in 2010. This caused a slight inconvenience to the 40 participants, but the meeting

will be remembered as a successful, albeit adventurous, event. For this meeting,

we are pleased to see that we have 60 registered participants. We have 6 invited

speakers, 3 workshops and 35 lectures during this meeting.

Selected papers of this seminar will be published in a special issue of Journal of

Hyperstructures according to the editorial policies of this journal.

Hossein Abdolzadeh

Jafar Azami

Adel P. Kazemi

Ahmad Khojali

Ahmad Yousefian Darani

Ardabil, August 26, 2012

Extended Abstracts of the

Second Seminar on Algebra and its Applications

September 13, 2012

Contents

1 PROGRAM OF SEMINAR 5

2 INVITED SPEAKERS 10

On Graphs Attached to Commutative Rings 11

A. Badawi

Hypercubes: Problems and Applications 12

E. S. Mahmoodian

On the Ratliff-Rush Closure of Ideals 13

R. Naghipour

Zero-divisor Graph of Content Algebras 17

P. Nasehpour

The Aluffi Algebra of a Singular Hypersurface 18

A. Nasrollah Nejad

Powers of Stanley-Reisner Ideals 19

S. Yassemi

3 WORKSHOPS 20

Zero-divisor Graphs of Nilpotent-free Semigroups 21

P. Nasehpour

1

Algebraic Aspects of Blowups 22

A. Nasrollah Nejad

Study of Maple Software and Group Theory Package 23

N. Rezaie Melal∗ and M. Ghaemi

4 LECTURES 25

Novikov Structures on Lie Algebras 27

R. Abdolmaleki∗ and H. Sharifi

Novikov Products of Derivations 28

R. Abdolmaleki∗ and H. Sharifi

On a Class of Deficiency Zero Groups 31

H. Abdolzadeh

Filtered Ring Derived from Discrete Valuation Ring and its Prop-erties 32

M. H. Anjom Shoa∗ and M. H . Hosseini

Unconditioned Strong d-sequence Relation with Weakly Proreg-ular Sequence and its Properties 33

M. H. Anjom Shoa

Conditions for the Existence of Self-dual Extended TransitivePermutation Codes of Finite Groups 34

A. Bandehbahman

On The Spectrum of Classical Prime Subhypermodules 38

S. Ebrahimian∗, S. Fathiye and A. Yousefian Darani

On non Abelian Tensor Analogues of Left Engel Elements ofGroups 42

E. Mohammadzadeh∗ and H. Golmakani

The Marginal Subgroup of Nilpotent Product of Cyclic Groups 45

A. Hokmabadi and F. Mohammadzadeh∗

On Almost Prime Subsemimodules 49

Farkhondeh Farzalipour

2

On n-absorbing Submodules 51

Farkhondeh Farzalipour and Fatemeh Farzalipour∗

Classical Prime Subhypermodules 53

S. Fathiye∗, S. Ebrahimian and A. Yousefian Darani

Direct Methods for Solving Systems of Linear Interval Equations 57

J. Garloff

On 2-absorbing Hyperideals of Multiplicative Hyperrings 58

P. Ghiasvand

On Artinian and semi-Artinian Modules 60

M. Gorbanalizadeh∗ and J. Azami

Linear Subspace Arrangement and its Applications 62

H. Haghighi

Two-sided Cayley Graphs: A New Generalization of Cayley Graphs 67

M. N. Iradmusa∗ and C. E. Praeger

” Total Dominating + Proper Coloring” in Graphs 68

A. P. Kazemi

Application of Laurent Series Ring in Wireless Communication 69

D. Kiani and H. Khodaiemehr∗

Completely Irreducible Submodules and Some Characterizationsof Distributive Modules 73

A. Khojali

Proof of Cantor’s Hypothesis 77

J. Kurdics

Invariant Finsler Metrics on Lie Groups 79

D. Latifi

Simple Power Graphs of Semigroups 82

S. Mohamadikhah∗ and N. V. Moosavi

On Graded Classical Prime Submodules 83

S. Motmaen

3

A Survey of k-tuple Total Domination in Graphs 87A. P. Kazemi and B. Pahlavsay∗

On Minimax and Artinian Modules 89N. Rafi’ei∗ and J. Azami

The Expansion Functor on a Bipartite Graph 91R. Rahmati-Asghar

Invariant Randers Metrics on Homogeneous Riemannian Mani-fold and Lie Group 96

Z. Ranjbar∗ and D. Latifi

Generalized Vandermonde Polynomial 100E. Rezaie

A note on quasikernels of the irreducible characters in finite p-groups 104

A. Saeidi

Multiplication Ideals in Γ-rings 106A. A. Estaji, A. Saghafi Khorasani∗ and S. Baghdari

2-absorbing Submodules of Multiplication Modules 107F. Soheilnia

Ideals of Novikov Algebras on Lie Algebras 108H. Tayer

Leibniz Algebras Admitting non–degenerated Derivations 109C. Zargeh

Total Coloring of Powers and Fractional Powers of a Graph 113M. s. Najafian, M. Zavieh∗

4

1 PROGRAM OF SEMINAR

5

Thursday, 30 August 2012

17:00 Registration

18:00 Opening

19:00 The Aluffi Algebra of a Singular HypersurfaceA. Nasrollah Nejad

Friday, 31 August 20128:00 - 8:50 On Graphs Attached to Commutative Rings

Ayman Badawi

9:00 - 9:50 Zero-divisor graph of content algebrasPeyman Nasehpour

9:50-10:20 Coffee Break

Class A Class B Class C Class D10:20-10:45 Direct Methods for Solving

Systems of Linear Interval

Equations

On a class of deficiency

zero groups

On 2-absorbing Hyperide-

als of Multiplicative Hy-

perrings

Completely irreducible

Submodules and Some

Characterizations of

Distributive Modules

J. Garloff H.Abdolzadeh P.Ghiasvand A. Khojali

10:50-11:15 Invariant Finsler metrics

on Lie group

Conditions for the Ex-

istence of Self-dual Ex-

tended Transitive Permu-

tation Codes of Finite

Groups

Linear subspace arrange-

ment and its applications

Linear subspace arrange-

ment and its applications

D. Latifi A.Bandehbahman S. Fathiye H. Haghighi

11:20-11:45 Novikov Products of

Derivations

Simple Power graphs of

semigroups

Unconditioned Strong

d-sequence Relation

with Weakly Proreg-

ular Sequence and its

Properties

On The Spectrum of Clas-

sical Prime Subhypermod-

ules

R. Abdolmaleki S.Mohamadikhah M. H. Anjom Shoa S. Ebrahimian

11:50-12:15 Total coloring of powers

and fractional powers of a

graph

On non Abelian Tensor

Analogues of Left Engel

Elements of Groups

Generalized Vandermonde

Polynomial

On Almost Prime Sub-

semimodules

M. Zavieh E.Mohammadzadeh E. Rezaei Farkhondeh Farzalipour

12:20 Group Photo (In front of the Faculty)

12:30-14:30 Lunch

Class A Class B Class C Class D14:30-14:55 Invariant Randers metrics

on homogeneous Rieman-

nian manifold and Lie

group

A note on quasikernels of

the irreducible characters

Multiplication ideals in Γ-

rings

A Survey of k-tuple Total

Domination in Graphs

Z. Ranjbar A. Saeidi A. S. Khorasani B. Pahlavsay

15:00-15:25 Novikov Structures on Lie

Algebras

On graded classical prime

submodules

2-absorbing Submodules of

Multiplication Modules

On n-absorbing Submod-

ules

R. Abdolmaleki S. Motmaen F. Soheilnia Fatemeh Farzalipour

Class A Class B Class C15:30-16:20 Workshop 1 Workshop 2 Workshop 3


16:50-17:15 ” Total Dominating +

Proper Coloring” in

Graphs

On Artinian and semi-

Artinian modules

The expansion functor on

a bipartite graph

The expansion functor on

a bipartite graph

A. P.Kazemi M.Ghorbanalizadeh N. Rafiei R. Rahmati-Asghar

17:30 Excursion (SPA, Sarein)

After Excursion Dinner

Saturday, 1 September 20128:00 - 8:50 Hypercubes: Problems and applications

Ebadollah S. Mahmoodian

9:00 - 9:50 ON the Ratliff-Rush closure of idealsReza Naghipour


Class A Class B Class C10:20-11:10 Workshop 1 Workshop 2 Workshop 3

11:20-11:45 Two-sided Cayley graphs: A new

generalization of Cayley graphs

The Marginal Subgroup of Nilpo-

tent Product of Cyclic Groups

Filtered Ring Derived from Discrete

Valuation Ring and its Properties

M. N. Iradmusa F. Mohammadzahed M. H. Anjom Shoa

11:50-12:15 Ideals of Novikov algebras on Lie al-

gebras

Leibniz algebras admitting non de-

generated derivations

Application of Laurent Series Ring

in Wireless Communication

H. Tayer C. Zargeh H. Khodaiemehr

12:30 Closing

13:00 Lunch

2 INVITED SPEAKERS

10


August 30–September 01, 2012, University of Mohaghegh Ardabili

On Graphs Attached to Commutative Rings

A. Badawi1

Abstract

Let R be a commutative ring with nonzero identity, and let Z(R) beits set of zero-divisors. In this talk, we discuss (briefly) various typesof graphs attached to the ring R, (i.e. zero-divisor graph and its gener-alization). In particular, the generalized total graph of the ring R willbe studied extensively. The zero-divisor graph of R is the (undirected)graph with vertices Z(R) − 0, and two distinct vertices x and y areadjacent if and only if xy = 0. The total graph of R is the (undirected)graph with vertices all elements of R, and two distinct vertices x and yare adjacent if and only if x+ y ∈ Z(R). In this talk, we will focus on ageneralization of the total graph of the ring R.

1Department of Mathematics, Faculty of Science, American University of Sharjah, Shar-jah, UAE, [email protected]

11



Hypercubes: Problems and Applications

E. S. Mahmoodian1

Abstract

Hypercubes are very interesting objects which arise in many differentareas of mathematics including Algebra, Analysis, Combinatorics andetc. As a graph, a hypercube Qn is a graph in which the vertices are allbinary vectors of length n, and two vertices are adjacent if and only ifthe Hamming distance between them is 1, i.e. their components differin exactly one place. Hypercubes have many applications and there aremany challenging conjectures about them. In this talk we discuss someof these conjectures and applications. Our emphasis will be on squarecoloring, tree packing, galactic number, and forced matchings.

1Department of Mathematical Sciences, Sharif University of Technology,[email protected]

12



On the Ratliff-Rush Closure of Ideals

R. Naghipour1

This talk will describe some interesting results about Ratliff-Rush closure ofideals. The important idea of the integral closure of an ideal in a commuta-tive Noetherian ring R (with non-zero identity) was introduced by Northcottand Rees [10]. This concept has been extended to ideals in an arbitrary com-mutative ring by L.J. Ratliff and D. Rush. Namely, the important notion ofRatliff-Rush closure of an ideal in a commutative ring R was introduced andstudied in [11] and [12] as a refinement of the integral closure of an ideal, andthis new idea has been proved useful in several questions, for example see [4],[5], [7], [14]. It is appropriate for us to provide a brief review. Let R be acommutative ring (with nonzero identity) and let I be an ideal of R. In [12]the interesting ideal,

I :=⋃

n∈N(In+1 :R In) = x ∈ R | xIn ⊆ In+1 for some n ≥ 1

of R, associated with I, has introduced by Ratliff and Rush. If grade I > 0,then this new ideal has some nice properties, for instance,

(0.1) for all sufficiently large In = In.

The Ratliff-Rush closure of an ideal I has been studied in [3], [4], [5], [6], [7],[13], [14] and has led to some interesting results.For an arbitrary non-zero finitely generated module M over a commutativeNoetherian ring R, we define the Ratliff-Rush closure I(M) of an ideal I of Rwith respect to M and we show that if grade (I,M) > 0, then

I(M) ⊇ I2(M)

⊇ · · · ⊇ In(M)

= InM :R M for all large n.

Also, we will obtain a finiteness result about the asymptotic prime divisors,

namely it is shown that for any ideal I of R the sequence AssRR/In(M)

n≥1 ofassociated primes is increasing and eventually stabilizes. Finally, we show that

whenever I and J are ideals of R such that I ⊆ J ⊆ I(M)a and grade (I,M) > 0,

1Department of Mathematics, University of Tabriz, Tabriz, Iran; and School of Mathemat-ics, Institute for Research in Fundamental Sciences (IPM), P.O. Box. 19395-5746, Tehran,Iran, [email protected]

13

then I(M) = J (M). Furthermore, if R is local and I is a Ratliff-Rush reductionof J with respect to M , then I contains a minimal Ratliff-Rush reduction Kof J and every minimal basis of K can be extended to a minimal basis of Iand the operation I → I(M) is a c∗-operation on the set of ideals I of R, (see,[1]). These results are further extensions of Mirbagheri-Ratliff-Rush’s resultsin [8] and [13].

In [4], a regular ideal I, i.e., grade I > 0, for which I = I is called a Ratliff-Rush ideal, and the ideal I is called the Ratliff-Rush ideal associated withthe regular ideal I. Subsequently, W. Heinzer et al. [6] introduced a conceptanalogous to this for modules over a commutative ring. Let us recall thefollowing definition:

Definition 1. (see, Heinzer et al. [6]). Let R be a commutative ring, letM be an R-module and let I be an ideal of R. The Ratliff-Rush closure of Iw.r.t. M , denoted by ˜IM , is defined to be the union of (In+1M :M In), wheren varies in N, i.e., ˜IM = e ∈M : Ine ⊆ In+1M for some n ∈ N.If M = R, then the definition reduces to that of the usual Ratliff-Rush idealassociated to I in R (see, [12]). Furthermore, ˜IM is a submodule of M , andit is easy to see that IM ⊆ IM ⊆ ˜IM . The ideal I is said to be Ratliff-Rushclosed w.r.t. M if and only if IM = ˜IM .

At the end of [6], the authors ask: What conditions ensure that all suitably highpowers of I are Ratliff-Rush closed w.r.t. M. That is: When does the abovecondition (0.1) extend to Ratliff-Rush closure with respect to a module? Thisis answered in this paper.

Let R be a Noetherian ring and M a finitely generated R-module. For anyideal I of R, we denote by GR(I) (resp. GM (I)) the associated graded ring⊕n≥0I

n/In+1 (resp. the associated graded GR(I)-module ⊕n≥0InM/In+1M).

W. Heinzer et al. have shown in ([6, Fact 9] that there exists an element inthe homogeneous ideal ⊕n≥1I

n/In+1 that is a non-zerodivisor on the moduleGM (I) if and only if for all positive integers n, ˜InM = InM . As a main resultof this paper, we characterize, when all powers of an ideal I are Ratliff- Rushclosed with respect to M in terms of the associated prime ideals of GM (I).More precisely we shall prove the following result:

Theorem 2. (see, [9]) Let R be a commutative Noetherian ring, let M bea non-zero finitely generated R-module, and let I be an M -proper ideal of Rsuch that grade (I,M) > 0. Then the following conditions are equivalent:

(i) All powers of I are Ratliff-Rush closed w.r.t. M .

(ii) For all p ∈ AssRM /uM , It * p.

(iii) There exists an integer k ≥ 1 and an element x ∈ Ik such that (In+kM :M

14

x) = InM for all integers n ≥ 1.

Throughout this paper, all rings considered will be commutative and will havenon-zero identity elements. Such a ring will be denoted by R, and the ter-minology is, in general, the same as that in [2]. Let I be an ideal of R, andlet M be a non-zero finitely generated module over R. We denote by R theRees ring R[u, It] := ⊕n∈ZI

ntn of R w.r.t. I, where t is an indeterminateand u = t−1. Also, the graded Rees module M [u, It] := ⊕n∈ZI

nM over Ris denoted by M , which is a finitely generated graded R-module. We shallsay that I is M -proper if M/IM 6= 0, and, when this is the case, we definethe M -grade of I (written grade (I,M)) to be the maximum length of all M -sequences contained in I. For any ideal I of R, the radical of I, denoted byRad(I), is defined to be the set x ∈ R : xn ∈ I for some n ∈ N.

References

[1] J. Amjadi and R. Naghipour, Asymptotic primes of Ratliff-Rush closure of ideals

with respect to modules, Comm. Algebra, 36, no. 5 (2008), 1942-1953.

[2] W. Bruns and J. Herzog, Cohen- Macaulay rings, Cambridge Univ. Press, Cam-

bridge, Uk, 1998.

[3] M. D anna, A. Guerrieri and W. Heinzer, Invariant of ideals having principal

reductions, comm. Algebra,29 (2001), 889-906.

[4] W. Heinzer, D. Lantz and K. Shah, The Ratliff-Rush ideals in a Noetherian ring,

Comm. Algebra 20 (1992), 591-622.

[5] W. Heinzer, B. Johnston, D. Lantz and K. Shah, The Ratliff-Rush ideals in a

Noetherian ring: A survey, in methods in module theory, pp. 149-159, Dekker, New

York, 1992.

[6] W. Heinzer, B. Johnston, D. Lantz and K. Shah, Coefficient ideals in and blowups

of a commutative Noetherian domain, J. Algebra 162 (1993) 355–391.

[7] A. V. Jayanthan and J. K. Verma, Local cohomology modules of bigraded Rees

algebras, Advances in algebra and geometry (Hyderabad, 2001), 39–52, Hindustan

Book Agency, New Delhi, 2003.

[8] A. Mirbagheri and L. J. Ratliff, Jr., On the relevant transform and the relevant

component of an ideal, J. Algebra 111 (1987), 507-519.

[9] R. Naghipour, Ratliff-Rush closures of ideals with respect to a Noetherian module,

J. Pure and Appl. Algebra, 195 (2005), 167-172.

15

[10] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge

Philos. Soc. 50 (1954), 145–158.

[11] L. J. Ratliff, Jr., ∆-closures of ideals and rings, Trans. Amer. Math. Soc. 313

(1989), 221–247.

[12] L. J. Ratliff, Jr. and D. Rush, Two notes on reductions of ideals, Indiana Univ.

Math. J. 27 (1978), 929-934.

[13] L. J. Ratliff, Jr. and D. Rush, ∆-reductions of modules, Comm. Algebra 21

(1993), 2667–2685.

[14] M. Rossi and I. Swanson, Notes on the behavior of the Ratliff-Rush filtration,

Contemp. Math 331 Amer. Math. Soc., Providence, RI, 2003.

16



Zero-divisor Graph of Content Algebras

P. Nasehpour1

In this talk, we prove that in content extensions minimal primes extend tominimal primes and discuss zero-divisors of a content algebra over a ring whohas Property (A) or whose set of zero-divisors is a finite union of prime ideals.We also examine the preservation of diameter of zero-divisor graph undercontent extensions.

1School of Mathematics, Institute for Research in Fundamental Sciences (IPM), [email protected]

17



The Aluffi Algebra of a Singular Hypersurface

A. Nasrollah Nejad1

Abstract

P. Aluffi to describe characteristic cycle in intersection theory intro-duced an intermediate graded algebra between the symmetric and theRees algebra of an ideal in a commutative Neotherin ring. These alge-bras investigated by A. Nasrollah Nejad and A. Simis. who named themAluffi algebras. A pair of ideals J ⊂ I of a commutative ring R satisfieslinearity condition if the symmetric algebra is isomorphic with the Aluffialgebra of I/J . In this talk, we give some necessary and sufficient condi-tion for linearity condition of a singular projective hypersurface. We areable to show that the singular locus of the generic member of a family ofquartic plane curves with isolated singularity satisfies linearity condition.We prove that some family of quintic and sextic singular plane curves ,thefixing singularity type, satisfies linearity condition. This work based onjoint work with Rashid Zaare Nahandi.

References

[1] P. Aluffi, Shadows of blow-up algebras, Tohoku Math. J. 56 (2004), 593-619.

[2] W. Fulton, Intersection Theory, Springer-Verlag, Berlin, 1984.

[3] A. Nasrollah Nejad, The Aluffi algebra of an ideal, PhD Thesis, Universidade

Federal de Pernambuco, Brazil, 2010.

[4] A. Nasrollah Nejad and R. Zaare Nahandi, Aluffi Torsion-free ideals, J. Algebra,

346 (2011) 284-298.

[5] A. Nasrollah Nejad and A. Simis, The Aluffi algebra, Journal of Singularities, 3

(2011) 20-47.

1Institute for Advanced Studies in Basic Sciences(IASBS) P. O. Box 45195-1159 Zanjan45137-66731 Iran, [email protected]

18



Powers of Stanley-Reisner Ideals

S. Yassemi1

Let ∆ be a simplicial complex and I∆ be the Stanley-Reisner ideal of ∆.Recently, there are several attempts to give precise combinatorial conditionsfor some algebraic behavior of the power (resp. symbolic power) of I∆.In this talk we present a survey on some research on (symbolic) power of theStanley-Reisner ideals.

1Department of Mathematics, University of Tehran, and School of Mathematics, Institutefor Research in Fundamental Sciences (IPM), [email protected]

19

3 WORKSHOPS

20



Zero-divisor Graphs of Nilpotent-free Semigroups

P. Nasehpour1

In this workshop, we find strong relationships between the zero-divisor graphsof apparently disparate kinds of nilpotent-free semigroups by introducing thenotion of an Armendariz map between such semigroups, which preserves manygraph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph.Then we give relationships between the zero-divisor graphs of certain topolog-ical spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtainingsurprisingly strong structure theorems relating ring-theoretic and topologicalproperties to graph-theoretic invariants of the corresponding graphs.

1School of Mathematics, Institute for Research in Fundamental Sciences (IPM), [email protected]

21



Algebraic Aspects of Blowups

A. Nasrollah Nejad1

The aim of this short course is to look at certain commutative graded alge-bras that appear frequently in algebraic geometry. One is here more inclinedtowards that algebra related to blowup and Resolution of Singularities, socalled Rees Algebra and their close associates as associated graded algebra,special fiber and Aluffi algebra. The Latter is a new graded algebra to describecharacteristic cycle parallel to Conormal cycles in intersection theory.

1Institute for Advanced Studies in Basic Sciences(IASBS) P. O. Box 45195-1159 Zanjan45137-66731 Iran, [email protected]

22



Study of Maple Software and Group TheoryPackage

N. Rezaie Melal∗1 and M. Ghaemi2

Abstract

In recent years, researchers have interested in mathematics software.Maple is one of the most widely used software that are highly regardedresearchers and scientists of mathematics. In this workshop, learn aboutthe work of software Maple, Maple software packages to include Alge-bra and GroupTheory are also studied. In the end, by using examples,examine the various directions will be discussed.

1 introduction

When the Maple project was conceived in November 1980 at the Universityof Waterloo, the primary goal was to design a computer algebra system thatwould be accessible to a large number of researchers in mathematics, engineer-ing, and science, and to a large number of students for educational purposes.The system would manipulate mathematics symbolically in a manner thatmaintained explicit algebraic form and where parameters remained variableat each stage of formulation. One could apply the rules of mathematics tosolving problems, thus deriving exact, closed-form solutions, without resort-ing to approximate, numerical techniques.Over the years, Maplesoft, in collaboration with many university researchlabs around the world, developed a large body of mathematical knowledge.State-of-the-art numeric libraries were integrated into Maple, extending itscapabilities beyond pure symbolics. Maple includes an extensive range of over3,500 functions. Many more packages are available as add-ons, extensions, orsimple source files. The Maple programming language is extremely powerfuland natural to use for developing technical programs that use math.The group package in Maple provides commands for working with finite groups

1Department of Mathematics, P. O. Box 45195, Zanjan, Iran, [email protected] of Mathematics, P. O. Box 45195, Zanjan, Iran, [email protected]

23

generated by permutations, and for groups defined by finite presentations bygenerators and defining relators. Permutation groups are constructed by us-ing the permgroup command. Certain commands (e.g., LCS) in the grouppackage apply only to finite permutation groups. Finitely presented groupsare created by using the grelgroup command.The members of each group can be identified with the multiplication tables.when the group order is large, it is impossible without using a computer. Dis-covery and Characterization of subgroups of a group that is the main contentsof group theory in this case is difficult. Therefore, specifying a set of genera-tors and a set of equations between the groups generators (called relations) tofacilitate members of the group and its subgroups. The algorithms developedin this field is the use of computer makes possible the require devaluations.

References

[1] W. Dongming, Software Tools and Applications, Imperial College Press, (2003).

[2] W. T. Hungerford, Algebra, Springer - Verlag, (1973).

24

4 LECTURES

25



Novikov structures on Lie algebras

Reza Abdolmaleki1 and Hesam Sharifi2

Abstract

In this paper we study Novikov algebras and Novikov structures onLie algebras. We show that semisimple algebras and reductive algebrascan be decompsed to subalgebras which all admit Novikov structures.

References

[1] B. Bakalov and V. Kac Field algebras, Int. Math. Res. Not, 3(2003), 123-159.

[2] A. A. Balanskii and S.P. Novikov, Poisson brackets of hydrodynamic type, Ferobe-

nius algebras and Lie algebras, Sov. Math. Dokl, 32(1985), 228-231.

[3] D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics,

Central Eur. j. Math, 4(2006), 323-357.

[4] D. Burde, K. Dekimpe, Novikov structures on solvable Lie algebras, J. Geom. Phys.

Not, 56(9)(2006), 1837-1855.

[5] R. Dijkgraaf, Chiral deformations of conformal field theories, Nucl. Phys. B,

499(3)(1997), 588-612.

[6] I. Frenkel, Y. Z. Huang and L. Lepowsky, On axomatic approaches to vertex oper-

ator algebras and modules, Mem. Am. Math. Soc, 494(1993), 1-64.

[7] J. E. Humphreys, Introduction to Lie algebras and Representation theory, Springer.

New York. appl, (1972),

[8] M. A. Semenov-Tian-Shansky, What is a classical r-matrix, Funct. Anal. appl,

17(1983), 259-272.

1Department of Mathematics, Faculty of Scince, Shahed University, Tehren, Iran,[email protected]

2Department of Mathematics, Faculty of Scince, Shahed University, Tehren, Iran, [email protected]

27



Novikov Products of Derivations

R. Abdolmaleki∗1 and H. Sharifi2

Abstract

If (A, .) is an associative and commutative algebra and D is a deriva-tion of A, for all x, y in A, x ·D(y) is a Novikov product. In this paperwe show that underlying Lie algebra of this Novikov product can bedecomposed to two solvable ideals.

Keywords: Novikov algebra, Novikov structure, derivation.

1 Introduction

Left-symmetric algebras or pre-Lie algebras first have been introduced byCayley in 1896. A Novikov algebra is a special case of a left-symmetric algebra.Novikov algebras have important states in mathematics and physics. Theyare appear in connected with Vertex algebras, conformal fields, Hamiltonianoperators, poisson brackets of hydrodynamic type and several other contecxts.

In this paper we study Novikov structures arise from derivations.

2 Theorems and lemmas

An algebra (A, ·) over k with product (x, y) 7→ x · y is called left-symmetricalgebra (LSA), if the product is left-symmetric, i.e., if the identity

x · (y · z)− (x · y) · z = y · (x · z)− (y · x) · z

is satisfied for all x, y, z ∈ A.

1Department of Mathematics, Faculty of Science, Shahed University, Tehren, Iran,[email protected]

2Department of Mathematics, Faculty of Science, Shahed University, Tehren, Iran, [email protected]

28

The algebra is called Novikov, if in addition

(x · y) · z = (x · z) · y

is satisfied.

Definition 1. An affine structure on a Lie algebra L over F is a left-symmetricproduct L× L→ L satisfying

[x, y] = x · y − y · x

for all x, y, z ∈ L. If the product is Novikov, we say that L admits a Novikovstructure.

Proposition 2. Any finite-dimensional Lie algebra admitting a Novikovstructure is solvable[3].

Proposition 3. Let V be a finite-dimensional vector space space over afield F and x ∈ End(V ). There exist unique xn, xs ∈ End(V ) satisfyingthe conditions: x = xn + xs, xs is semisimple, xn is nilpotent, xn and xscommute[7].

Lemma 4. Let A be a finite-dimensional F-algebra. then DerA contains thesemisimple and nilpotent parts of all its elements.

Proposition 5. Let (A, ·) be an associative and commutative algebra andD be a derivation of A. Then the product x y = x · D(y) is Novikov. Inparticular, it defines a Novikov structure on the Lie algebra L given by

[x, y] := x y − y x = x ·D(y)− y ·D(x).

3 New results

Result 6. Lie algebra of proposition 3, is decomposed to two solvable ideal.

Result 7. Let D be non-semisimple and non-nilpotent derivation of algebraA. Then three different Lie algebras arise from D.

References

29

[1] B. Bakalov and V. Kac Field algebras, Int. Math. Res. Not, 3(2003), 123-159.

[2] A. A. Balanskii and S.P. Novikov, Poisson brackets of hydrodynamic type, Ferobe-

nius algebras and Lie algebras, Sov. Math. Dokl, 32(1985), 228-231.

[3] D. Burde, Classical r-matrices and Novikov algebras, Geom. Dedicate, 122(1)(2006),

145-157.

[4] D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics,

Central Eur. j. Math, 4(2006), 323-357.

[5] R. Dijkgraaf, Chiral deformations of conformal field theories, Nucl. Phys. B,

499(3)(1997), 588-612.

[6] I. Frenkel, Y. Z. Huang and L. Lepowsky, On axomatic approaches to vertex oper-

ator algebras and modules, Mem. Am. Math. Soc, 494(1993), 1-64.

[7] J. E. Humphreys, Introduction to Lie algebras and Representation theory, Springer.

New York. appl, (1972),

[8] M. A. Semenov-Tian-Shansky, What is a classical r-matrix, Funct. Anal. appl,

17(1983), 259-272.

30



On a Class of Deficiency Zero Groups

H. Abdolzadeh1

Abstract

In this note we study the family 〈x, y | xl = yl, xyxmyn = 1〉 ofgroups, where l, m and n are positive integers. Certain finite subclasseshave been identified and the orders is calculated.

References

[1] G. Baumslag, J. W. Morgan and P. B. Shalen, Generalized triangle groups, Math-

ematical Proc.Cambridge.Soc. 102(1987), 25-31.

[2] M. J. Beetham and C. M. Campbell, A note on the Todd Coxeter coset enumeration

algorithm, Proc.Edinburgh Math.Soc., 20(1976), 73-79.

[3] H. S. M. Coxeter, The abstract groups Gm,n,p, Trans.Amer.Math.Soc., 45(1939),

73-150.

[4] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups,

4th edition, Springer-Verlag, Berlin/New York, 1984.

[5] H. Doostie and A. R. Jamali, A class of deficiency zero soluble groups of derived

length 4, Proc.Roy.Soc.Edinburgh , section A, 121(1992), 163-168.

[6] M. Edjvet and R. M. Thomas, The (l,m | n, k)-groups, J.Pure and Applied Alge-

bra, 114(1997), 175-208.

[7] J. Howie and R. M. Thomas, The groups (2 , 3 , p ; q); asphericity and a conjecture

of Coxeter, J. Algebra, 154(1993), 289-309.

[8] D. L. Johnson, Presentations of groups, London Math.Soc. Student Texts, 15Cam-

bridge University Press, Cambridge, 1990.

1Department of Mathematics and Application, University of Mohaghegh Ardabili, P. O.Box 179, Ardabil, Iran, [email protected]

31



Filtered Ring Derived from Discrete Valuation Ringand its Properties

M. H. Anjom Shoa∗1 and M. H. Hosseini2

Abstract

In this talk we show that if R is a discrete valuation ring then R isa filtered ring and we prove some properties and relations when R is adiscrete valuation ring.

1Department Of Mathematics, Birjand University , Iran,, [email protected] Of Mathematics, Birjand University, Iran, [email protected]

32



Unconditioned Strong d-sequence Relation withWeakly Proregular Sequence and its Properties

M. H. Anjom Shoa1

Abstract

In this paper we obtain a relation between Unconditioned strong d-sequence and weakly proregular sequence in a commutative ring. Thenwe derive some new properties of unconditioned strong d-sequence fromthis relation.

References

[1] W. Bruns, J.Herzog, Cohen Macaulay rings, 39, Cambridge Univ.Press, Cam-

bridge,1993.

[2] N. Burbaki, Commutative Algebra, Chapter 1-7, Speringer-Verlag, Berlin, 1989.

[3] M. Brodmann, R. Sharp , Local Cohomology: An Algebraic Introduction with

Geometric Application, Cambridge Univ.Press , Cambridge 1998.

[4] T. D. Hamilton, T. Marley, Non-Noetherian Cohen macaulay Rings, Journal of

Algebra 307 (2007), 342-360.

[5] C. Huneke, M. Katzman, R. Y. Sharp, Y. Yao, Frobenius test exponent for param-

eter ideals in generalized Cohen Macaulay local rings, Journal of Algebra 305 (2006),

516-539.

[6] H. Matsumura, Commutative Ring Theory, Cambridge Univ.Press, Cambridge

1986.

[7] P. Schenzel, Proregular sequence, local cohomology , and completion, Math.Scand.

92(2)(2003), 271-289.

1Department Of Mathematics, Birjand University, Iran, [email protected], [email protected]

33



Conditions for the Existence of Self-dual ExtendedTransitive Permutation Codes of Finite Groups

A. Bandehbahman1

Abstract

We discuss on the condition which imply the existence of self-dualextended transitive permutation codes, when the underlying group hasodd order.

1 Introduction

Let F be a finite field of order q which is a prime power, and let X be a finiteset; by FX we denote the F -vector space with the basis X and with the usualscalar product. Any subspace C of FX is said to be a linear code over F .The subspace C⊥ which contains all vectors orthogonal to C. A linear code Cis said to be self-orthogonal if C ⊆ C⊥, and self-dual if C = C⊥. Further, ifX is multiplicative group,then FX is an algebra with multiplication inducedby the multiplication of the group X, and any ideal C of the algebra FX, iscalled a group code of group X over the field F . Coding researchers attractto obtain the conditions for the existence of self-dual group codes becauseself-dual codes with long length are good codes.Let G be a finite group and X be finite G-set. Extending the G-action on X

linearly, FX becomes an FG-permutation module. If C is an FG-submoduleof FG-permutation module of FX, C is called permutation code of FX.From module theory, we know that FG-module V , the dual space V ∗ :=HomF (V, F ) which denotes the F -space of all linear forms on V , becomes anFG-module in natural way, for g ∈ G and γ ∈ V ∗, gγ(v) = γ(g−1v) for allv ∈ V , the FG-module V ∗ is called the dual module of V . We say that V isself-dual module if V ∼= V ∗. Let f be a bilinear form on FG-module V andU be a submodule of V . We called that U is hyperbolic submodule if

U = U⊥ := v ∈ V | f(u, v) = 0 ∀u ∈ U.

If V has a hyperbolic submodule then we say that V is a hyperbolic FG-module. If U ⊆ U⊥ (equivalently, the restriction of f on U is zero) then we

1Department of Mathematics, University of Isfahan, Isfahan, Irananahita [email protected]

34

say that U is an isotropic submodule.In [4], some condition for the existence or non-existence of self-dual permuta-tion codes are obtained. Assume that X is finite group, and extend the set Xto X which union of X with a single point set, then the vector space FX isa module over the algebra FX and any ideal C of FX is called an extendedgroup code. In Sect. 2 we present new condition for existence of self-dualextended transitive permutation codes, by [6].

2 Main result

Let X = X⋃x0 be the extended G-set, where x0 /∈ X and x0 is G-fixed.

Assume C is permutation submodule of FX, the permutation code C is calledextended permutation code of X over F , denoted by C 6 FX.

Definition 1. We say that C is a extended transitive permutation codeof X over F , if X be a transitive G-set.

Definition 2. Let m, t ∈ N with gcd(m, t) = 1. We denote by s = st(m) thesmallest nonnegative integer such that

m | ts − 1.

Furthermore, for a finite group G we put

st(G) = lcmst(p) | p | |G|, p a prime and p 6= t.

Lemma 3. ( [3]) Let X be a transitive G-set. If the characteristic q of F isprime to the order of G, then the trivial FG-module F appears in FX exactlyonce.

Theorem 4. ( [4]) Assume V is FG-module. If any composition factor of Vis not self-dual, then V is hyperbolic.

Theorem 5. ([1]) If X be a transitive G-set, then FX = IndGGx(F ) where

Gx is stabilizer of x in G.

Theorem 6. ([4]) Let G be a finite group, and let X be a transitive G-setwhere n := |X| is coprime to |F |. The following two are equivalent:

1. There is permutation code C of FX such that C⊥ = C ⊕ F and −n isa square element of F .

35

2. There is a self-dual permutation code C of FX.

Theorem 7. Let G be a finite group of odd order, and let X be a transitiveG-set. If F algebraically close field and characteristic of F is prime to theorder of G, then there exists a transitive permutation code C of FX such thatC⊥ = C ⊕ F.

As consequence of Theorem 7 and Theorem 6, we have the following at once.

Theorem 8. Let notation be as above. If −n is a square element of F , thenthere is a self-dual extended permutation code of X over F .

Theorem 9. Assume that F is even characteristic. Let G be a finite group ofodd order, and let X be a transitive G-set. If char(F ) - |G| and (log2(F ))2 >s2(G), then there exists a transitive permutation code C of FX such thatC⊥ = C ⊕ F.

Theorem 10. Let notation be as in Theorem 9. If −n is a square element ofF , then there is a self-dual extended transitive permutation code of X over F .

References

[1] J. L. Alperin and R. B. Bell, Groups and Representations, Springer-Verlag ,

Berlin, 1995.

[2] F. Bernhardt, P. Landrock,and O. Manz, The extended Golay codes considered

as ideals, J. Comb. Theory, 55 (1990), 234-246.

[3] Y. Fan and Y. Yuan, On Self-dual Permutation Codes, Acta Math. Sci. Ser. B,

28 (2008), 633-638.

[4] Y. Fan and G. Zhang, On the existence of self-dual permutation codes of finite

groups, Des. Codes Cryptography, 62 (2012), 19-29.

[5] B. Huppert, Character Theory of Finite Groups II, Walter de Gruyter, Berlin,

1998.

[6] M. Loukaki, Hyperbolic modules and cyclic subgroups, J. Algebra, 266 (2003),

34-50.

[7] C. Martinez-Perez, W. Willems, Self-dual extended cyclic codes, Appl. Algebra

Eng. Com. Comp., 17 (2006), 1-16.

36

[8] S. Roman, Coding and Information Theory, Springer-Verlag, New York, 1992.

[9] W.Willems, A note on self-dual group code, IEEE Trans. Info. Theory, 48 (2002),

3107-3109.

37



On the Spectrum of Classical PrimeSubhypermodules

S. Ebrahimian∗, S. Fathiye and A. Yousefian Darani1

Abstract

Let R be a commutative hyperring with identity and let M be an R-hypermodule. We topologize Cl.Hspec(M), the collection of all classicalprime subhypermodules of M , with the Zariski topology and prove somebasic results.

1 Preliminaries

The study of algebraic hyperstructures is a well established branch of classi-cal algebraic theory. Hyperstructure theory was born in 1934 [3] when Martydefined hypergroups, began to analysis their properties and applied them togroups, rational functions and algebraic functions. Now, they are widely stud-ied from the theoretical point of view and for their applications to many sub-jects of pure and applied mathematics nowadays, for example, polygroupswhich are certain subclasses of hypergroups are used to study color algebra.A comprehensive review of the theory of hyperstructures appears in [1, 2, 4,5].A hyperstructure is a non-empty setH together with a mapping “” : H×H →P ∗(H), where P ∗(H) is the set of all the non-empty subsets of H.If x ∈ H and A,B ∈ P ∗(H), then by A B, A x and x B, we meanA B =

⋃a∈Ab∈B

a b, A x = A x and x B = x B, respectively.

Now,we call a hyperstructure (H, ) a hypergroup if the following axioms aresatisfied:(i) for every x, y, z ∈ H,x (y z) = (x y) z;(ii) for every x, y ∈ H,x y = y x;(iii) there exists a 0 ∈ H such that 0 x = x, for all x ∈ H;(iv) for every x ∈ H, there exists a unique element x′ ∈ H such that 0 ∈ xx′.(we call the element x′ the opposite of x).

1Department of Mathematics and Applications, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran, [email protected], [email protected] [email protected]

38

A hyperring is an algebraic structure (R,+, ·) which satisfies the followingaxioms:

(1) (R,+) is a hypergroup; (we shall write −x for x′ ).

(2) (R, ·) is a semigroup having zero as a bilaterally absorbing element.

(3) The multiplication is distributive with respect to the hyperoperation “+”.

The following facts follow easily from the above axioms:

−(−x) = x and −(x+ y) = −x− y.

A hyperring (R,+, .) is called commutative with identity 1 ∈ R, if we have

a) xy = yx, for all x, y ∈ R;

b) 1x = x, for all x ∈ R.

A subhyperring I of a hyperring R is a left (resp., right) hyperideal of Rprovided that rx ∈ I, (resp., xr ∈ I,) for all r ∈ R, and x ∈ I. I is called ahyperideal if I is both left and right hyperideal and in this case it is denotedby I R. A hyperideal I of a hyperring R is said to be prime if for everyr, s ∈ R, rs ⊆ I implies that either r ∈ I or s ∈ I.

A non-empty setM is called a left hypermodule over a hyperring R (or simplyan R-hypermodule) if (M,+) is a hypergroup and there exists a map “ · ” :R×M → P ∗(M) by (r,m) 7→ r.m such that for all r1, r2 ∈ R andm1,m2 ∈M ,we have

(i) r1.(m1 +m2) = r1.m1 + r1.m2;

(ii) (r1 + r2).m1 = r1.m1 + r2.m1;

(iii) (r1r2).m1 = r1(r2.m1)

A non-empty subset N of a hypermodule M is said to be a subhypermoduleif (N,+, ·) is itself a hypermodule. In this case we write N ≤ M . Let N be aproper subhypermodule of M . N is said to be a prime subhypermodule of M ,whenever rm ⊆ N with r ∈ R and m ∈ M \ N , implies that rM ⊆ N . Theprime spectrum Hspec(M) of an R-hypermodule M is defined to be the set ofall prime subhypermodules of M . If N ≤M is an R-subhypermodule, denoteby V (N) the variety of N , which is the set consisting of all prime subhyper-modules of M that contain N . Note that for any family of subhypermodulesNi(i ∈ I) of M ,

⋂i∈I V (Ni) = V (

∑i∈I Ni). Thus if ξ(M) denotes the col-

lection of all subsets V (N) of Hspec(M), then ξ(M) contains the empty setand Hspec(M), and ξ(M) is closed under arbitrary intersections. We shallsay that M is a hypermodule with the Zariski topology, or a top-hypermodulefor short, if ξ(M) is closed under finite unions, for in this case ξ(M) satisfiesthe axioms for the closed subsets of a topological space.

In what follows, all hyperrings are commutative hyperrings with identity and

39

all hypermodules are considered left hypermodules.

2 Main results

LetR be a commutative hyperring with identity and letM be anR-hypermodule.In this section we introduce the concept of classical prime subhypermodulesand topologize Cl.Hspec(M), the collection of all classical prime subhyper-modules of M , with the Zariski topology and prove some basic results.

Definition 1. Let R be a commutative hyperring with identity. A propersubhypermodule N of an R-hypermodule M is said to be classical prime, ifwhenever rsm ⊆ N with r, s ∈ R and m ∈ M \ N , then either rm ⊆ N orsm ⊆ N .

The set of all classical prime subhypermodules of M is called the classicalprime hyperspectrum of M and is denoted by CL.Hspec(M). Note that theCL.Hspec(M) may be empty for some hypermodule M . A hypermodule issaid to be primeless if CL.Hspec(M) = ∅. A subhypermodule N of an M iscalled semiprime (resp., classical semiprime) if N is an intersection of prime(resp., classical prime) subhypermodules of M . Also a prime (resp., classicalprime) subhypermodule P of M is called extraordinary (resp., classical ex-traordinary) if whenever N and L are semiprime (resp., classical semiprime)subhypermodules of M with N

⋂L ⊆ P , then either N ⊆ P or L ⊆ P .

For any classical prime subhypermodule S of M , V(S) denotes the set of allclassical prime subhypermodules of M containing S, that is V(S) = p ∈CL.Hspec(M) | S ⊆ p.

Theorem 2. Let M be an R-hypermodule. The following statements hold:

(i) V(M) = ∅ and V(0) = CL.Hspec(M),

(ii)⋂

i∈I V(Ni) = V(∑

i∈I Ni) for every family Nii∈I of subhypermodules ofM,

(iii) V(N1)⋃

V(N2) ⊆ V(N1⋂N2), for every subhypermodules N1 and N2 of

M.

Thus if η(M) denotes the collection of all subsets V(N) of CL.Hspec(M),then η(M) contains the empty set and CL.Hspec(M) and is closed underarbitrary intersection. If also η(M) is closed under finite unions, i.e for anysubhypermodules N1 and N2 of M there exists a subhypermodule N3 of Msuch that V(N1)

⋃V(N2) = V(N3), then η(M) satisfies the axioms for the

closed subsets of a topological space. We call this the Zariski topology. A

40

hypermodule with this Zariski topology is called a classical Top-hypermodule.Among the other results we prove:

Theorem 3. Let R be a commutative hyperring with identity, and M anR-hypermodule. Then, the following statements are equivalent:

(i) M is a classical Top-hypermodule.

(ii) Every classical prime subhypermodule of M is classical extraordinary.

(iii) V(N) ∪ V(L) = V(N ∩ L) for any classical semiprime subhypermodulesN and L of M .

Theorem 4. Every classical Top-hypermodule is a Top-hypermodule.

References

[1] P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Advances in

Mathematics (Dordrecht), Kluwer Academic Publishers, Dordrecht, 2003.

[2] B. Davvaz, A brief survey of the theory of Hv-structures, Algebraic hyperstructures

and applications (Alexandroupoli-Orestiada, 2002), 39–70, Spanidis, Xanthi, 2003.

[3] F. Marty, Sur une generalization de la notation de grouse 8th Congress, Math

Scandianaves, Stockholm, (1934),45-49.

[4] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press Inc.,

Palm Harber, USA (1994).

[5] M. Zahedi, M. Ameri R, On the prime, primary and maximal subhypermodules,

Italian Journal of Pure and Applied Mathematics, 5(1999), 61-80.

41



On non Abelian Tensor Analogues of Left EngelElements of Groups

E. Mohammadzadeh∗1 and H. Golmakani2

Abstract

Tensor analogues of n-Engel groups were introduced by Moravec. Hedescribed the structure of tensor analogues of 2-Engel groups. With thehelp of these results we investigate the properties of left tensor Engelelements. Moreover we prove that if x2 = 1 and G

⊗G is of exponent

2n, then x ∈ L⊗n+1(G).

1 Introduction

For a group G the nonabelian tensor square G⊗G is a group generated by

the symbols g⊗h, subject to the relations

gg′ ⊗

h = (gg′ ⊗

hg′)(g

′ ⊗h) and g

⊗hh

′= (g

⊗h

′)(gh

′ ⊗hh

′),

Where g, g′, h, h

′ ∈ G and gh = h−1gh. Also tensor analogues of right n-Engelelements have been defined. Recall that the set of right n-Engel elements ofa group G is defined by Rn(G) = a ∈ G : [a,n x] = 1, forallx ∈ G. It wasshown that for n ≥ 3 the set Rn(G) is not necessarily a subgroup [2]. The setof rightn⊗-Engel elements of a group G is then defined as

R⊗n (G) = a ∈ G : [a,n−1 x]

⊗x = 1⊗, forallx ∈ G. Biddle and Kappe[3]

proved that R⊗2 (G) is always a characteristic subgroup of G containing Z(G)

and contained in R2(G). A group G is an n -Engel group if [x,n y] = 1 for allx, y ∈ G. The group G is said to be n⊗-Engel when [x,n−1 y]

⊗y = 1⊗ for

all x, y ∈ G. It is obvious that every n⊗-Engel group is also n-Engel. It ishardly surprising that the if G is a 2⊗-Engel group , then G

⊗G is abelian.

The next result is the key to striking analogies between the tensor annihilatorand the centralizer of a subset X in a group G.Proposition [4]:Let G be a group. Then there exists a homomorphism k :G⊗G −→ G

′such that g

⊗h −→ [g, h]. Furthermore kerk denoted by

1Department of Mathematics Payame Noor University, I. R. Iran , [email protected]

2Department of Mathematics, Faculty of science, Mashhad Branch, Islamic Azad Univer-sity, 91735-413, Iran, [email protected]

42

J2(G) is a central subgroup of G⊗G and G acts trivially on J2(G).

Note that Z⊗n (G) = a ∈ G : [a, x1, ..., xn−1]

⊗x = 1⊗, forallx1, ..., xn ∈ G

is a characteristic subgroup of G contained in the n-th center Zn(G). Thissubgroup is called the n-th tensor center of G.

Every 2⊗

-Engel group is also 2-Engel. Now we show that R⊗n (G) ⊆ Rn(G)

,L⊗n (G) ⊆ Ln(G). Also Abdollahi proved that if x2 = 1 and G

′is of exponent

dividing 2n then x ∈ Ln+1(G). Now we get a similar result on non abeliantensor square of groups.

2 Main Results

In this section we show that R⊗n (G) ⊆ Rn(G) ,L

⊗n (G) ⊆ Ln(G). Furthermore

we prove that If x ∈ L⊗3 (< x, xy >) then x ⊆ Z

⊗2 < x, xy > . Also we get the

result that if x2 = 1 and G⊗G is of exponent 2n, then x ∈ L

⊗n+1(G).

Definition 1. Let G be a group and x ∈ G. Then x is called left 3⊗

-Engelelement if [a, x, x]

⊗x = 1⊗, foralla ∈ G. The set of left 3⊗-Engel elements

of a group can be defined as L⊗3 (G) = x ∈ G : [a, x, x]

⊗x = 1⊗, foralla ∈

G.

Theorem 2. Let G be a group. Then

(i)R⊗n (G) ⊆ Rn(G).

(ii)L⊗n (G) ⊆ Ln(G).

Proof. (i) Let k : G⊗G −→ G

′be a homomorphism given by g

⊗h −→

[g, h].Let a ∈ R⊗n (G) and x ∈ G. Thenk([a,n−1 x]

⊗x) = [[a,n−1 x], x] =

1,hence a ∈ Rn(G). The inclusion L⊗n (G) ⊆ Ln(G) is proved in a similar way.

Theorem 3. Let G a group, and x ∈ G such that x2 = 1. Then

(i)[g,n x]⊗x = (g

⊗x)−2n for all g ∈ G and integers n ≥ 1.

(ii) If every g⊗x is a 2-element , then x ∈ L

⊗(G)for all g ∈ G. In particular

,if G⊗G is a 2-group ,then x ∈ L

⊗(G).

(iii) If every g⊗x is of order dividing 2n then x ∈ L

⊗n+1(G) for all g ∈ G. In

particular, if G⊗G is of exponent 2n, then x ∈ L

⊗n+1(G).

Proof. (i) We proceed by induction on n. Since x2 = 1,

1 = g⊗x−1x = (g

⊗x)(g

⊗x−1)x. Therefore (g

⊗x−1)x = (g

⊗x)−1. Now

n = 1 holds by lemma(1) Moravec. Now[g,n x]

⊗x = [[g,n−1 x], x]

⊗x

= ([g,n−1 x]⊗x)−1([g,n−1 x]

⊗x)x (by part (c) of lemma 1 Moravec)

43

= ((g⊗x)(−2)n−1

)−1(((g⊗x)(−2)n−1

)x)=((g

⊗x)(−2)n−1

)−1((g⊗x)−1)(−2)n−1

)= (g

⊗x)2∗2

n−1

= (g⊗x)−2n

= ((g⊗x)(−2)n−1

)−1

(ii) Let g⊗xis a 2-element. Then by part (i) we have [g, nx]

⊗x = (g

⊗x)(−2)n =

1, so we have the result.(iii) It is clear by part (ii).

Acknowledgements: The outhors thanks the research council of MashhadBranch(Islamic Azad University) for support.Also, we would like to thankreferee for his/her many helpful suggestions.

References

[1] P.Moravec On nonabelian tensor analogues of 2-Engel conditions (2007).

[2] Macdonald Some examples in the theory of groups, Mathematical Essays dedicated

to A. J. Maclntyre, Ohio UniversityPress (1970), 263-269.

[3] D. P. Biddle, L. C. Kappe , On subgroups related to the tensor center, Glasg.

Math. J. 45 (2003), 323-332.

[4] R. Brown, D. L. Johnson and E. F. Robertson Some computations of nonabelian

tensor product of groups, J. Algebra 111 (1987), 177-202.

44



The Marginal Subgroup of Nilpotent Product ofCyclic Groups

A. Hokmabadi 1 and F. Mohammadzadeh∗2

Abstract

Every nilpotent group is a homomorphic image of a nilpotent productof cyclic groups. In this paper, we give the structure of marginal subgroupof some nilpotent products of cyclic groups with respect to the variety ofpolynilpotent groups, under some conditions.

1 Introduction

Let Gii∈I be a family of cyclic groups, then the nth nilpotent product ofthe family Gii∈I is defined to be the group

n∗∏

i∈IGi =

∏∗i∈I Gi

γn+1(∏∗

i∈I Gi),

where∏∗

i∈I Gi is the free product of the family Gii∈I . If I is a finite set ,then the nth nilpotent product of cyclic groups G1, G2, . . . , Gt is also denoted

by G1n∗ G2

n∗ · · · n∗ Gt.

Every nilpotent group is a homomorphic image of a nilpotent product of cyclicgroups. So investigation on nilpotent products of cyclic groups can be a start-ing point to study nilpotent groups. A. Magidin [2,3] determined the structureof the center of this group with some conditions. Let Nc1,c2,...,cl be the vari-ety of polynilpotent groups. In this paper, we give the structure of marginal

subgroup of the nth nilpotent product G = Zn∗ ... n∗ Z︸︷︷︸

m−copies

n∗ Zα1

n∗ ... n∗ Zαr with

respect to the variety Nc1,c2,...,cl , when αi+1 divides αi for all i, 1 ≤ i ≤ r − 1,and (p, α1) = 1 for any prime p less than k + n and also l = 1 or c1 6= 1, inwhich k = c1 + c2(c1 + 1) + · · ·+ cl(cl−1 + 1) · · · (c1 + 1).

1Department of Mathematics,Faculty of Sciences, Payame Noor University, Iran,[email protected]

2Department of Mathematics,Faculty of Sciences, Payame Noor University, Iran,[email protected]

45

2 Main result

First we need to recall the following definitions.

Definition 1. If V is a variety of groups defined by a set of words V and Gis a group, then the marginal subgroup of G is defined to be

V ∗(G) = a ∈ G|v(g1, . . . , gia, . . . , gr) = v(g1, . . . , gr),for all v ∈ V , g1, . . . , gr ∈ G, 1 ≤ i ≤ r.

Definition 2. Let F be the free group with the set of free generators x1, x2, x3, . . ..The word γc+1 = [x1, x2, . . . , xc+1] defines the variety of all nilpotent groupsof class at most c. the variety of polynilpotent groups of class row (c1, . . . , cl)which is denoted by Nc1,c2,...,cl , is defined by polynilpotent words as follows:

γc1+1,c2+1,...,cl+1 = γc1+1 γc2+1 · · · γcl+1.

The following lemma is very useful to determine the marginal subgroup withrespect to the variety of polynilpotent groups.

Lemma 3. (Hekster [1]). Let V = Nc1,c2,...,cl be the variety of polynilpotentgroups of class row (c1, . . . , cl) and N be a normal subgroup of a group G.Then the following are equivalent:i) N ⊆ V ∗(G),ii) [N,c1 G,c2 γc1+1(G), . . . ,cl γc1+1,c2+1,...,cl−1+1(G)] = 1,where V ∗(G) is the marginal subgroup of G.

Let V = Nc1,c2,...,cl be the variety of polynilpotent groups of class row (c1, . . . , cl)such that l = 1 or c1 6= 1 and let k is the integer c1+c2(c1+1)+ · · ·+cl(cl−1+1) · · · (c1 + 1).Also, suppose that Ai = 〈xi〉 for 1 ≤ i ≤ r, and Bj = 〈yj〉 for 1 ≤ j ≤ m.yj is of infinite order and xi is of finite order αi such that αi+1 | αi, for all i,1 ≤ i ≤ r − 1.

Our purpose is determining the structure of marginal subgroup of G = B1n∗

· · · n∗ Bmn∗ A1

n∗ · · · n∗ Ar, when all the primes appearing in the factorizationof α1 are larger than k + n− 1.

It is easy to see that if n ≤ k then V ∗(G) = G. Therefore we study the casen > k. For this we use induction and the following lemma gives the first stepof the induction.

46

Lemma 4. Consider the above assumption and let H = B1k+1∗ · · · k+1∗

Bmk+1∗ A1

k+1∗ · · · k+1∗ Ar . If H is not cyclic, then

V ∗(H) =

〈γ2(H), xα21 〉 m = 0,

〈γ2(H), yα11 〉 m = 1,

γ2(H) m ≥ 2.

Proof. Suppose m = 0. Then Lemma 3 implies that 〈γ2(H), xα21 〉 ⊆ V ∗(H).

To prove the reverse inclusion, let w ∈ V ∗(H). Then w = xσ11 x

σ22 · · ·xσr

r c, suchthat c ∈ γ2(H) ⊆ V ∗(H). Using the following epimorphism:

θi : A1k+1∗ · · · k+1∗ Ar → A1

k+1∗ Ai

we can conclude that xσii ∈ V ∗(A1

k+1∗ Ai)∩Ai (2 ≤ i ≤ r). On the other hand,

applying [4, Lemma 1.3] and [4, Theorem 3], we have V ∗(A1k+1∗ Ai) ∩Ai = 1

for all i, 2 ≤ i ≤ r, and V ∗(A1k+1∗ A2) ∩ A1 = 〈xα2

1 〉. This completes theproof of the case m = 0.Suppose m = 1. we proceed as in the case m = 0 but we use the epimorphisms

θi : B1k+1∗ A1

k+1∗ · · · k+1∗ Ar → B1k+1∗ Ai,

for all i, 1 ≤ i ≤ r.For the case m ≥ 2, use the epimorphisms

θj : B1k+1∗ · · · k+1∗ Bm

k+1∗ A1k+1∗ · · · k+1∗ Ar → B1

k+1∗ Bj ,

for 2 ≤ j ≤ m, and the epimorphisms

ϕi : B1k+1∗ · · · k+1∗ Bm

k+1∗ A1k+1∗ · · · k+1∗ Ar → B1

k+1∗ Ai,

for 1 ≤ i ≤ r. 2

Now we are ready to prove the main result of the paper.

Theorem 5. Consider the above assumption and let H = B1k+n∗ · · · k+n∗

Bmk+n∗ A1

k+n∗ · · · k+n∗ Ar such that all the primes appearing in the factor-ization of α1 are larger than k + n − 1. Then V ∗(H) = H if H is cyclic, andotherwise

V ∗(H) =

〈γn+1(H), xα21 〉 m = 0,

〈γn+1(H), yα11 〉 m = 1,

γn+1(H) m ≥ 2.

47

Proof. We can assume that H is not cyclic. Now we proceed by induction onn. Lemma 2.4 gives the result for n = 1. Suppose that the result holds for

the (k + n− 1)th nilpotent product of Bi and Ai. If M = B1k+n−1∗ · · · k+n−1∗

Bmk+n−1∗ A1

k+n−1∗ · · · k+n−1∗ Ar, then by induction hypothesis

V ∗(M) =

〈γn(M), xα21 〉 m = 0,

〈γn(M), yα11 〉 m = 1,

γn(M) m ≥ 2.

On the other hand, M = H/γk+n(H). Then by considering the natural epi-morphism ψ : H →M , we have ψ(V ∗(H)) ⊆ V ∗(M). Therefore

V ∗(H) ⊆

〈γn(H), xα21 〉 m = 0,

〈γn(H), yα11 〉 m = 1,

γn(H) m ≥ 2.

Now suppose m = 0. Then applying [4, Lemma 1.3] and [4, Lemma H2], wecan prove that xα2

1 ∈ V ∗(H). Hence the inclusions 〈γn+1(H), xα21 〉 ⊆ V ∗(H) ⊆

〈γn(H), xα21 〉 hold. Let g ∈ V ∗(H). Then g = xtα2

1 c∏h

i=1 zδii , where t and δi are

integers, c ∈ γn+1(H) and z1, . . . , zh are basic commutators of weight exactlyn on x1, . . . , xr. Then one can show that

∏hi=1 z

δii = 1. we can continue this

process and use [4, Lemma 1.3] several times to conclude that zδii = 1, for all1 ≤ i ≤ h, and hence g ∈ 〈γn+1(H), xα2

1 〉. Therefore V ∗(H) = 〈γn+1(H), xα21 〉.

Proofs of the cases m = 1 and m ≥ 2 are similar to the previous case. 2

References

[1] N. S. Hekster, Varieties of groups and isologisms, J. Austral. Math. Soc., Ser. A,

46(1989), 22-60.

[2] A. Magidin, Capability of nilpotent product of cyclic groups, J. Group Theory,

8(4)(2005), 431-452.

[3] A. Magidin, Capability of nilpotent product of cyclic groups II, J. Group Theory,

10(4)(2007), 441-451.

[4] R. R. Struik, On nilpotent products of cyclic groups, Canad. J. Math., 12(1960),

447-462.

48



On Almost Prime Subsemimodules

F. Farzalipour1

Abstract

Let R be a commutative semiring and M an R-semimodule. Thispaper is concerned with generalizing some results of module theory tosemimodule theory. Here, we study almost prime subsemimodules ofsemimodules and we obtain some basic properties of almost prime sub-semimodules.By a commutative semiring we mean an algebraic system R = (R,+, .)such that R = (R,+) and R = (R, .) are commutative semigroup, con-nected by a(b+c) = ab+bc for all a, b, c ∈ R, and there exists 0 ∈ R suchthat r + 0 = 0 and r.0 = 0.r = 0 for all r ∈ R. Throughout this paperlet R be a commutative semiring. A semimodule M over a semiring Ris a commutative additive semigroup which has a zero element, togetherwith a mapping from R ×M into M such that (r + s)m = rm + sm,r(m+n) = rm+ rn, r(sm) = (rs)m and 0m = r0M = 0Mr = 0M for allm,n ∈ M and r, s ∈ R. A proper subsemimodule N of an R- semimod-ule M is called almost prime if rm ∈ N − (N : M)N where r ∈ R andm ∈M , then m ∈ N or r ∈ (N :M).

Theorem 1. Let M be a semimodule over a commutative semiring R and Nbe a proper subtractive (=k-)subsemimodule of M . The following are equiv-alent:(i) N is an almost prime subsemimodule.(ii) For r ∈ R− (N :M), (N :M (r)) = N ∪ ((N :R M)N :M (r)),(iii) For r ∈ R − (N : M), (N :M (r)) = N or (N :M (r)) = ((N :R M)N :M(r)).

Theorem 2. Let N be a k-subsemimodule of R-semimodule M and K aQ-subsemimodule of M with K ⊆ N . Then if N is an almost prime subsemi-module of M , then N/K is an almost prime subsemimodule of M/K.

Theorem 3. Let M be an R-semimodule and S a multiplicatively closedsubset of R. Then if N is an almost prime subsemimodule of M , then NS isan almost prime subsemimodule of RS-module MS .

1Department of Mathematics, Payame Noor University, Tehran 19395-3697, Iran. e-mail:f [email protected]

49

Theorem 4. Let M be a multiplication R-semimodule and N be a properk-subsemimodule of M . Then N is an almost prime subsemimodule of M ifand if only if for any ideal I of R and subsemimodule K of M with IK ⊆ N ,we have I ⊆ (N :M) or K ⊆ N

Theorem 5. LetM be a finitely generated faithful multiplicationR-semimoduleand N a proper k-subsemimodule of M . Then the following are equivalent:(i) N is an almost prime subsemimodule of M .(ii) (N:M) is an almost prime ideal of R.(iii) N = QM for some almost prime ideal Q of R.

References

[1] P. J. Allen, Ideal theory in semirings, Dissertation Texas Christian Univesity, 1969.

[1] F. Farzalipour and P. Ghiasvand, On almost prime ideals in semirings, Journal of

Advanced Research in Pure Math., (2012), to appear.

[2] H. A. Khashan, On almost prime submodules, Acta Mathematica Scientia., 32(2012),

645-651.

50



On n-absorbing Submodules

Farkhondeh Farzalipour1 and Fatemeh Farzalipour∗2

Abstract

Let R be a commutative ring with nonzero identity and let M bea unitary R-module. In this article we investigate some properties ofn-absorbing submodules of an R-module. A proper submodule N ofan R- module M is called n-absorbing if whenever a1...anm ∈ N fora1, ..., an ∈ R and m ∈ M , then a1...an ∈ (N : M) or there are n − 1 ofai

,s whose produced with m is in N .n-absorbing ideals in commutative rings have been studied by D. F. An-derson and A. Badawi ([1]). A. Yousefian Darani and F. Soheilnia ([4])have defined the concept of n-absorbing submodules and generalized someproperties of n-absorbing ideals to n-absorbing submodules. Now, wewant to study a number of results concerning such submodules.

Theorem 1. LetM be a cyclic faithful R-module and N a proper submoduleof M . Then the following are equivalent:(i) N is an n-absorbing ssubmodule of M .(ii) (N:M) is an n-absorbing ideal of R.(iii) N = QM for some n-absorbing ideal Q of R.

Theorem 2. Let P1, P2, ..., Pn be prime submodules of a cyclic faithful R-module M such that (Pi : M) are pairwise comaximal. Then P1P2...Pn is ann-absorbing submodule of M .

Theorem 3. Let N be an n-absorbing submodule of M and S be a multi-plicatively closed subset of R.(i) If S−1N 6= S−1M , then S−1N is an n-absorbing submodule of S−1M .(ii) If W is an n-absorbing submodule of S−1R-module S−1M , then W c is ann-absorbing submodule of M .

Theorem 4. Let R = R1×R2 where Ri, i = 1, 2, is a commutative ring withnonzero identity and let M =M1 ×M2 where Mi is an Ri-module. Then the

1Department of Mathematics, Payame Noor University, Tehran 19395-3697,f [email protected]

2Department of Mathematics and Application, University of Mohaghegh Ardabili, P. O.Box 179, Ardabil, Iran, [email protected]

51

following hold:(i) N1 is an n-absorbing submodule of M1 if and only if N1 ×M2 is an n-absorbing submodule of M .(ii) N2 is an n-absorbing submodule of M2 if and only if M1 × N2 is an n-absorbing submodule of M .

References

[1] D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings,

Comm. Algebra, 39(2011), 1647-1672.

[2] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Aust. Math. Soc. ,

75(2007), 412-429.

[3] A. Yousefian Darani and F. Soheilnia, 2-absorbing and weakly 2-absorbing submod-

ules, Thai J. Math. 9(2011), 577-584.

[4] A. Yousefian Darani and F. Soheilnia, On n-absorbing submodules, Mathematical

Communications, (2012), to appear.

52



Classical Prime Subhypermodules

S. Fathiye∗1 , S. Ebrahimian2 and A. Yousefian Darani3

Abstract

Let R be a commutative hyperring and M a hypermodule. A propersubhypermodule N of M is called a classical prime subhypermodule,whenever rsm ⊆ N with r, s ∈ R and m ∈ M , implies that rm ⊆ N orsm ⊆ N . In this talk we will study basice properties of classical primesubhypermodules and we shall compare classical prime subhypermoduleswith prime subhypermodules.

1 Introduction and Preliminaries

Hypergroups, were introduced about 70 years ago by Marty at the VIII Congressof Scandinavian Mathematicians (see [5]). Nowadays, hyperstructures have alot of applications to several domains of mathematics and computer science(see [2-6]) and they are studied in many countries across all the world. Onthe other hand, many researchers are engaged in extending the concepts ofabstract algebra, such as rings, modules, ideals, submodules, prime ideals, ...to the framework of the algebraic hyperstructuess (see for example [6, 8]).

Let us recall some notions and basic results about hyperstructures that weshall use in this paper.

Let H be a nonempty set and let P ∗(H) be the set of all nonempty subsetsof H. A hyperoperation on H is a map o : H ×H → P ∗(H). In this case thecouple (H, o) is called a hypergroupoid. If x ∈ H and A,B ∈ P ∗(H), then byA B, A x and x B, we mean

A B =⋃a∈Ab∈B

a b, A x = A x and x B = x B.

1Department of Mathematics and Applications, University of Mohaghegh Ardabili, P. O.Box 179, Ardabil, Iran, [email protected]



53

A hypergroupoid (H, o) is called a semihypergroup if for all x, y, z ∈ H wehave xo(yoz) = (xoy)oz. An element e ∈ H is called an identity of (H, o) if forevery a ∈ H, we have a ∈ (eoa)∩ (aoe). We say that a semihypergroup (H, o)is a hypergroup if for all x ∈ H, we have xoH = Hox = H. A subhypergroup(K, o) of the group (H, o) is a nonempty subset K of H, such that for allk ∈ K, we have koK = Kok = K.

The hypergroup (R,+) with internal composition ”.” is called a hyperring if

(i) (R, .) is a multiplicative semigroup having zero as a bilaterally absorbingelement;

(ii) z(x+ y) = zx+ zy, (x+ y)z = xz + yz, for all x, y, z ∈ R.

A hyperring (R,+, .) is called commutative whith identity 1 ∈ R, if we have

a) xy = yx, for all x, y ∈ R;

b) 1x = x1, for all x ∈ R.

A subhyperring I of a hyperring R is a left (right) hyperideal of R providedthat rx ∈ I(xr ∈ I) for all r ∈ R and x ∈ I. I is called a hyperideal if I isboth left and right hyperideal and in this case it is denoted by I R.

A (left) unitary R-hypermodule M over a hyperring R with identity is a hy-pergroup (M,+) with an external composition (r,m) −→ am from R×M toM satisfying the following conditions. For all r, s ∈ R and m,n ∈M ;

(i) r(m+ n) = rm+ rn;

(ii) (r +m)m = rm+ rn;

(iii) (rs)m = r(sm);

(iv) 1m = m and 0m = 0, for all r, s ∈ R and m.n ∈M .

A nonempty subset N of an R-hypermodule M is called a subhypermoduleif N is itself an R-hypermodule with the operations of M . We write thenN ≤M .

From now on all hyperrings are commutative with identity and all hypermod-ules are unitary.

Let N be a proper subhypermodule of M . N is said to be a prime subhyper-module of M , whenever rm ⊆ N with r ∈ R and m ∈ M \ N , implies thatrM ⊆ N . A hyperideal I in a hyperring R is said to be prime (primary) ifrs ⊆ I and r /∈ I implies that s ∈ I (sn ∈ I, for all n ∈ N) for r, s ∈ R.A proper subhypermodule N of an M is called maximal, provided that forK ≤ M with N ⊆ K ⊆ M , we have either K = N or N = M . It is notdifficult to show that a maximal subhypermodule is a prime subhypermodule(see [8]). Recall that a proper subhypermodule N of a hypermodule M is

54

said to be a primary subhypermodule if the condition rm ⊆ N, r ∈ R andm ∈M \N , implies that rnM ⊆ N , for some n ∈ N .

2 Classical prime subhypermodules

Let R be a commutative hyperring and M a hypermodule. A proper sub-hypermodule N of M is called a classical prime subhypermodule, wheneverrsm ⊆ N with r, s ∈ R and m ∈M , implies that rm ⊆ N or sm ⊆ N . Henceif we consider R as an R-hypermodule, then classical prime subhypermodulesare prime exactly hyperideals of R.

Theorem 1. Any prime subhypermodule is classical prime.

Recall that a subhypermodule N ofM is prime subhypermodule provided thatrm ⊆ N , with r ∈ R and m ∈M , implies that eitherm ∈ N or rM ⊆ N . Thisis equivalent to say that if I is an hyperideal of R and K is a hypersubmoduleof M with IK ⊆ N , then either K ⊆ N of IM ⊆ N . The next Theoremshows that the same result is true for classical prime subhypermodules.

Theorem 2. Let N be a subhypermodule of M . Then N is a classical primesubhypermodule of M if and only if whenever I and J are hyperideals of Rand K is a subhypermodule of M such that IJK ⊆ N , then either IK ⊆ Nor JK ⊆ N .

Lemma 3. LetN be a classical prime subhypermodule ofM . Then, (N :R M)is a prime hyperideal.

Clearly, every prime subhypermodule is classical prime and primary, but theconverse do not necessarily hold. But we have the next result:

Theorem 4. Let N be a subhypermodule of M . Then N is a prime subhy-permodule of M if and only if N is both primary and classical prime.

Theorem 5. Let M,N be hypermodules and P,Q be classical prime sub-hypermodules of M and N respectively. Assume that f : M −→ N is ahypermodule strongly homomorphism. Then(i) If f is onto and Kerf ⊆ P , then f(P ) is classical prime subhypermoduleof N .(ii) f−1(Q) is a classical prime subhypermodule of M .(iii) If f is onto, then there is a bijection between the set of all classical primesubhypermodules of M containing Kerf and the set of all classical prime

55

subhypermodules of N .

Theorem 6. Let Nii∈I be a family of classical prime subhypermodules ofR-hypermodule M, such that (Ni : M) = W C R for all i ∈ I. Then

⋂i∈I Ni

is a classical prime subhypermodule of M.

References

[1] P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Advances in

Mathematics (Dordrecht), Kluwer Academic Publishers, Dordrecht, 2003.

[2] B. Davvaz, A brief survey of the theory of Hv-structures, Algebraic hyperstructures

and applications (Alexandroupoli-Orestiada, 2002), 39–70, Spanidis, Xanthi, 2003.

[3] V. Leoreanu-Fotea, B. Davvaz, Join n-spaces and lattices, Multiple Valued Logic

Soft Comput. (15) (2008) (in press).

[4] V. Leoreanu-Fotea, B. Davvaz, n-Hypergroups and binary relations, European J.

Combin. 29 (2008) 1207-1218.

[5] F. Marty, Sur une generalization de la notation de grouse 8th Congress, Math

Scandianaves, Stockholm, (1934),45-49.

[6] C. Massouros, Free and Cyclic Hypermodules, Annali di Mathematica pure odap-

plicata, (IV), Vol. CL. (1988) 153-166.

[7] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press Inc.,

Palm Harber, USA (1994).

[8] M. Zahedi and M. Ameri, On the prime, primary and maximal subhypermodules,

Italian Journal of Pure and Applied Mathematics, 5 (1999), 61-80.

56



Direct Methods for Solving Systems of LinearInterval Equations

J. Garloff1

The topic of our talk are systems of linear interval equations, i.e., linear sys-tems where the coefficients of the matrix and the right hand side vary betweengiven bounds. We consider direct methods for the enclosure of the solutionset of such a system. The algorithms are obtained from the ordinary elimina-tion procedures by replacing the real numbers by the related intervals and thereal operations by the respective interval operations. We report on methodsby which the breakdown of the interval variants of the Gaussian eliminationcaused by division of an interval containing zero can be avoided for someclasses of matrices with identically signed inverses. The approach consists ofa tightening of the interval pivot by determining the exact range of the pivotover the matrix interval. By means of the interval Cholesky method an en-closure of the solution set for symmetric matrices can be found. We presenta method by which the diagonal entries of the interval Cholesky factor canbe tightened for positive definite interval matrices, such that a breakdown ofthe algorithm can be prevented. In the case of positive definite symmetricToeplitz matrices, a further tightening of the interval pivots and other entriesof the Cholesky factor is possible.

1University of Applied Sciences / HTWG Konstanz Faculty of Computer Scienceand University of Konstanz Department of Mathematics and Statistics Konstanz, Germany,Prof [email protected]

57



On 2-absorbing Hyperideals of MultiplicativeHyperrings

P. Ghiasvand1

Abstract

In this article, we define 2-absorbing hyperideals of multiplicative hy-perrings as a new generalization of prime hyperideals. We study someproperties of 2-absorbing hyperideals of multiplicative hyperrings andthen we obtain some related results. The notion of multiplicative hyper-ring introduced by R. Rota [4] in 1982. 2-absorbing ideals in commutativerings were discussed by A. Badawi in [2]. In this paper we generalizedthese notions to the hypercase and obtains several results.

A triple (R,+, o) is called a multiplicative hyperring if(1) (R,+) is an abelian group;(2) (R,o) is semihypergroup;(3) for all a, b, c ∈ R, we have ao(b+ c) ⊆ aob+ aoc and (b+ c)oa ⊆ boa+ coa;(4) for all a, b ∈ R, we have ao(−b) = (−a)ob = −(aob).A non empty subset I of a multiplicative hyperring R is a hyperideal if(1) a, b ∈ I, then a− b ∈ I;(2) If x ∈ I and r ∈ R, then rox ⊆ I.

Definition 1. A nonzero proper hyperideal P of a multiplicative hyperringR is called to be 2-absorbing if xoyoz ⊆ P where x, y, z ∈ R, then xoy ⊆ P oryoz ⊆ P or xoz ⊆ P .

Theorem 2. Let I be a 2-absorbing hyperideal of a multiplicative hyperringR. Then Rad(I) is a 2-absorbing hyperideal of R and x2 ⊆ I for every x ∈Rad(I).

Theorem 3. Let I be a 2-absorbing hyperideal of a multiplicative hyperringR. Then there are at most two prime hyperideals of R that are minimal overR.

1Department of Mathematics, Payame Noor University, Tehran 19395-3697, Iran,p [email protected]

58

Theorem 4. Let I is a 2-absorbing hyperideal of multiplicative hyperring R.Then one of the following statements must hold:(i) Rad(I) = P is a prime hyperideal of R such that P 2 ⊆ I.(ii) Rad(I) = P1 ∩ P2, P1P2 ⊆ I, and Rad(I)2 ⊆ I where P1, P2 are the onlydistinct prime hyperideals of R that minimal over I.

Theorem 5. Suppose that I is a P -primary hyperideal of a multiplicativehyperring R. Then I is a 2-absorbing hyperideal of R if and only if P 2 ⊆ I. Inparticular, M2 is a 2-absorbing hyperideal of R for each maximal hyperidealM of R.

Theorem 6. Let P is a nonzero divided prime hyperideal of R and I is ahyperideal of R such that Rad(I) = P . Then the following statements areequivalent:(i) I is a 2-absorbing hyperideal of R.(ii) I is a P -primary hyperideal of R such that P 2 ⊆ I.

References

[1] D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings,

Comm. Algebra, 39(2011), 1647-1672.

[2] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Aust. Math. Soc.,

75(2007), 412-429.

[3] P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Aca-

demic Publishers, (2003)

[4] R. Rota, Sugli iperanelli moltiplicativi, Rend. Di Math., Series VII, 4(1982), 711-

724.

[4] M. N. Zahedi and R. Ameri, On the prime, primary and maximal subhypermodules,

Italian Journal of pure and Applied Math., 5(1999), 61-81.

59



On Artinian and semi-Artinian modules

M. Gorbanalizadeh∗1 and J. Azami2

Abstract

Let R be a commutative Noetherian ring and let M be a unitaryR-module. In this paper we investigate some new properties about min-imax, semi-Artinian, Noetherian and Artinian R-modules.

Definition 1. An R-module M is called radical, if it has no maximal sub-module. By P (M) we denote the sum of the radical submodule of M . P (M)is the largest radical submodule of M . If P (M) = 0, M is called reduced.

Definition 2. An R-moduleM is called minimax if it has a finitely generatedsubmodule N such that M/N is Artinian.

Definition 3. A monomorphism f : M → N of R-modules is said to beessential if Imf be an essential submodule of N .

Definition 4. An R-module M is called semi-Artinian if every proper sub-module contains a minimal submodule. We denote by L(M) the sum of allArtinian submodule of M . L(M) is the largest semi-Artinian R-module andalways has a decomposition L(M) = ⊕m∈Max(R)Lm(M), where Lm(M) =Σ∞n=1(0 :M mn) and Max(R) is the set of all maximal ideal of R.

Definition 5. An R-module M is called strongly faithful, if rM Artinianimplies r = 0 for any r ∈ R. In general a submodule of a strongly module isnot strongly faithful.

Definition 6. Let M be an R-module. Then Coass(M)= p ∈ Spec(R) |P is the annihilator of an Artinian factor module of M

Theorem 7. Let R be a Noetherian ring and M be a strongly faithfulR−module. Let K and N be R-submodules of M such that K is Artinian

1Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ard-abil, Iran, [email protected]


60

R-module. Let N be a maximal element of the set X ⊂ M | X ∩ K = 0.Then the natural homomorphism f : K →M/N is monomorphism and essen-tial. Also N is strongly faithful.

Theorem 8. Let M be an R-module such that LM) is reduced and Artinian.Then L(M) is Noetherian.

Theorem 9. Let M be an R-module which is minimax and semi Artinian.Then M is Artinian.

Theorem 10. Let M be an R-module and I be an ideal of a Noetherian ringR such that 0 :M I is Artinian(Noetherian) R-module. Then for all positiveinteger n the R-module 0 :M In is Artinian(Noetherian) respectively.The following theorem is proved in [3]. We give a new proof for this theorem

Theorem 11. Let (R,m) be a complete Noetherian local domain. For anR-module M if 0 ∈ Coass(M), then there is an epimorphismM → E :=E(R/m). Also If M be a torsion-free then R can be embedded in M .

Proof. Since 0 ∈ Coass(M), then there is a submodule N of M such that0 :R M/N = 0 and M/N is Artinian module. Since 0 :R M/N = 0, it followsthat 0 :R HomR(M/N,E) = 0. On the other hand HomR(M/N,E) is finitelygenerated and so we can write HomR(M/N,E) = (x1, ...xt). Therefore 0 =0 :R HomR(M/N,E) = ∩t

i=1(0 :R xi)and consequently there exist 1 ≤ i ≤ tsuch that 0 :R xi = 0. Now R can be embedded in HomR(M/N,E) and sothere is an epimorphism from M to E which complete the proof.

References

[1] E. Matlis, Injective modules over Noetherian rings, Pacific J. Math, 8(1958), 511-

528.

[2] E. Matlis, 1-Dimensional Cohen-Macaulay Rings, Lecture Notes in Mathematics,

Springer, Berlin, 327(1973).

[3] P. Rudlof, On minimax and related modules, Can. J. Math, 44(1)(1992), 154-166.

61



Linear Subspace Arrangement and its Applications

H. Haghighi1

A linear subspace arrangement in an (n + 1)-dimensional vector spaceV over a field K is a finite collection A = V1, V2, . . . , Vs of affine (linear)subspaces of V , such that non of the element of A is contained in the other one.

While geometric objects such as: Grassmannian G(r, n), the variety of r di-mensional linear subspaces of Pn, scrolls, ruled surfaces or more generally vec-tor bundles, higher dimensional secant varieties of a variety and many otherexamples parameterizes an infinite family of linear subspaces of a projectivespace and much of their properties are studied extensively in algebraic geom-etry, the finite union of linear subspaces in a projective space or in an affinespace are less studied and still less of their properties are known. For examplethere are still no general results about the Hilbert functions of a finite unionof linear spaces in Pn, or the generators of a finite union of linear spaces in aprojective space or an affine space.

Subspace arrangements have been studied from different point of views. Specif-ically one can name the following ones:

(1) Commutative Algebra:

(a) The generators of ideal of W = V1∪V2∪· · ·∪Vs. Even though eachVi is generated by linear polynomial, but the generators of I(W ) arenot in general in the form of product of linear polynomial. A majorproblem in this field is to determine the arrangements A which thedefining ideal of the union of the elements of A is generated by theproduct of linear polynomials [8].

(b) Hilbert polynomial of I(W ). It can be shown in certain cases, thispolynomial is a combinatorial invariant.

(c) Castelnuovo– Mumford Regularity of an ideal (or module) is a mea-sure for its complexity. For linear subspace arrangement A in Pn

the Castelnuovo–Mumford regularity of its defining ideal is equal

1K. N. Toosi University of Technology, Mathematics Department, Tehran, Iran,[email protected]

62

to s. This immeadiately implies that the minimal generators ofdefining ideal of the subspace arrangement has degree at most s.

(d) Free resolution of ideal of subspace arrangement. In particular de-termining the cases where this resolution is linear [9].

(2) Combinatorics:

(a) The primary decomposition of edge ideal of simple graph G de-termines a subspace arrangement in An where n = ]V (G), byVi = Zeros(P ). A major problem is to determine those graphswhich dimVi = dimVj for every pair of associated prime ideals ofedge ideal. Another related problem is to characterize those oneswhich their ideal of definition is Cohen-Macaulay.

(b) Let Vi = Zeros(xi) for each coordinate function in An. Then thedefining ideal of V1∪· · ·∪Vn is x1x2 . . . xn. A lattice will be assignedto this hyperplane which is isomorphic to power set of 1, . . . , n.An antichain in this lattice generate an abstract simplicial com-plex. Conversely every simplicial complex on n vertices determinean antichain and hence a linear subspace arrangement.

(c) Let G be a simple graph. To each vertex v ∈ V (G) one can assigna line Lv which is isomorphic to P1. Then by embedding theselines in a projective space and in this space these projective linesdetermine a subspace arrangement called graph curve.

(3) Algebraic Geometry:

(a) Hartshorne and Hirschowitz proved that the Hilbert polynomial ofr generic lines L1, . . . , Lr in PN is min

(N+nn

), r(n + 1). When

dimLi > 1 it is not known whether the similar formula is hold forHilbert Function of generic linear subspace arrangement [6].

(b) There is a classical problem which asks whether exists a uniquerational normal curve in Pn which passes through n+3 points whichare in general position in Pn. In recent years this problem has beengeneralized to linear subspace arrangement, that is the question iswhether there exists a rational normal curve which passes throughp points and intersecting ` codimension two linear spaces in n − 1points each. The problem has a solution if p+ ` = n+3 and linearsubspaces and points are in general positions in Pn [2].

(c) A nice application of linear subspace arrangement is Waring’s prob-lem for homogeneous polynomial of degree d in n+ 1 variable [3].

63

(4) Algebraic Topology:

(a) The homotopy type of complement of a subspace arrangement isstudied [5].

(b) For K = R, the number of regions of the complements of linearsubspace arrangement V1, . . . , Vs, that is, Rn − ∪s

1Vi is a combi-natorial invariant.

(5) Applications:

(a) A specific application of subspace arrangement is in statistics inGeneralized Principal Component Analysis. The main problem isto approximate a collection of data in a vector space, by a collec-tion of linear spaces. In disciplines such machine learning, computervision, robotics, the problem of estimating a collection of modelsfrom sample data points is very essential. There is an importantrelation between algebraic properties of subspace arrangements andsubspace segmentation problem. In fact an important class of in-variants are given by Hilbert functions of subspace arrangements[7].

(b) In topological robotics, in planning motion of n particles withoutcollision, the complement of a subspace arrangement will be usedto compute the complexity of the motion [4].

1 Basic definitions and examples

Let V1, . . . , Vs be linear subspace of an n-dimensional vector space V over afield K. The variety W = V1 ∪ V2 ∪ · · · ∪ Vs is called a subspace arrangementin V .

Example 1. The ideal

〈(x1 − x2) . . . (x1 − xn)(x2 − x3) . . . (x3 − xn) . . . (xn−1 − xn)〉 ⊂ K[x1, . . . , xn]

defines a subspace arrangement Vij | 1 ≤ i < j ≤ n in Kn where eachsubspace Vij is a hyperplane in Kn defined by the equation xi − xj = 0. Thissubspace arrangement is known as braid arrangement.Since V ∼= Kn we can identify each Vi with its isomorphic copy in Kn. Thenthe ideal of W is I(W ) = I(V1) ∩ · · · ∩ I(Vs). Since

√I(V1) . . . I(Vs) =

√I(V1) ∩ · · · ∩ I(Vs) = I(V1) ∩ · · · ∩ I(Vs) = I(W ),

64

I(W ) would be defining ideal of W . Even though the ideals I(V1) . . . I(Vs)and I(V1) ∩ · · · ∩ I(Vs) are equal up to radical, but each one can shed morelight into the properties of the subspace arrangement. For example, the idealI(V1) . . . I(Vs) is generated in degree s and its Hilbert series has a close con-nection with the Betti numbers of this ideal.

Example 2. Let E = e1, . . . , en be the standard basis of the vector spaceKn. Let V1, . . . , Vs be a linear subspace arrangement of Kn, such that eachVi has a subset of E as its basis. Then the ideal of definition of W = V1 ∪· · · ∪ Vs is generated by square free monomials and in fact it is the Stanley-Reisner ideal of the simplicial complex ∆ on 1, . . . , n generated by Fi = k |ek is a basis element of Vi.Since each I(Vi) is generated by linear forms, it is expected that I(W ) is alsogenerated by product of linear forms. But the following examples shows thatthis is not true in general.

Example 3. Let V1 = Z(x, z), V2 = Z(y, t) and V3 := Z(x + t, y + z), thenthe ideal of the subspace arrangement V1 ∪ V2 ∪ V3 in K4 is:

(yz − xt, z2t+ zt2, xzt+ xt2, x2t+ xyt, x2y + xy2),

and one of generators of I, that is yz − xt is not the product of linear formswhich define each Vi, i = 1, 2, 3.

In the sequel, by constructing specific examples, we describe how the itemsin the introduction can be illustrated by subspace arrangement. Moreoverwe explain some existing conjectures which naturally arise in the course ofstudying these type of objects.

References[1] A. Bjoner, Subspace arrangement, First European Congress of Mathematics (Paris

1992) , Vol. I. Progress in Mathematics 119 (Birkhauser, Basel, 1994) 321–370.

[2] E. Carlini and M. V. Catalisano, Existence results for rational normal curves, J.

London Math. Soc. (2) 76 (2007) 73–86.

[3] E. Carlini, M. V. Catalisano, and A. V. Geramita, Subspace arrangement, config-

uration of linear spaces and the quadratics containing them , Preprint.

[4] M. Farber and S. Yuzvinsky Topological robotics: Subspace arrangement and col-

lision free motion planning , Preprint.

65

[5] J. Grbic and S. Theriault, it The homotopy type of the complement of a coordinate

subspace arrangement, Preprint.

[6] R. Hartshorne and A. Hirschowitz, Droites en position general dans l’espace pro-

jectif. In Algebraic Geometry ( La Rabida, 1981) Volume 961 of Lecture Notes in

Mathematics, pages 169–188, Springer-Verlag.

[7] Y. Ma, A. Y. Yang, H. Derksen and R. Fossum Estimation of Subspace Arrange-

ments with Applications in Modeling and Segmenting Mixed Data, SIAM REVIEW,

Vol. 50, No. 3, pp. 413-45, 2008.

[8] J. Sidman, Defining equations of subspace arrangements embedded in reflection

arrangement, Int. Math. res. Not., 15:713–727, 2004.

[9] J. Sidman, Resolution and subspace arrangement, In Syzygies and Hilbert functions,

Volume 254 of Lecture Notes in Pure and Applied Math., pages 249–265, Chapman

and Hall/CRC, Boca Raton, FL. 2007.

66



Two-sided Cayley Graphs: A New Generalization ofCayley Graphs

M. N. Iradmusa∗1 C. E. Praeger2

Abstract

The class of Cayley graphs is an important subset of graphs. Thereare many advantages in applying Cayley graphs as models for intercon-nection networks. For example because of their vertex transitivity, wecan implement the same routing and communication schemes at eachnode of the network they model. But we know that this class does notcontain all vertex transitive graphs. Therefore, there have been variousgeneralizations of Cayley graphs, prototypes of transitive graphs, to findnew constructions of vertex transitive graphs specially non-Cayley vertextransitive graphs. In this paper, a new generalization of Cayley graphswill be introduced which is called Two-sided Cayley graphs. Also we willshow some of their properties and applications.

References

[1] G. Gauyacq, On quasi-Cayley graphs, Discrete Appl. Math. 77(1997), no. 1,

43-58.

[2] M. A. Iranmanesh and Cheryl E. Praeger, On non-Cayley vertex-transitive graphs

of order a product of three primes, J. Combin. Theory Ser. B 81(2001), no. 1, 1-19.

[3] A. V. Kelarev and C. E. Praeger, On transitive Cayley graphs of groups and

semigroups, European J. Combin. 24(2003), no. 1, 59-72.

[4] D. Marusic, Cayley properties of vertex symmetric graphs, Ars Combin. 16(1983),

no. B, 297-302.

[5] B. D. McKay and C. E. Praeger, Vertex-transitive graphs which are not Cayley

graphs. I, J. Austral. Math. Soc. Ser. A 56(1994), no. 1, 53-63.

[6] B. D. McKay and C. E. Praeger, Vertex-transitive graphs that are not Cayley

graphs. II, J. Graph Theory 22(1996), no. 4, 321-334.

1Department of Mathematical Sciences, Sharif University of Technology and School ofMathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran, [email protected]

2School of Mathematics and Statistics, The University of Western Australia, 35 StirlingHighway, Crawley, WA 6009, Australia, [email protected]

67



”Total Dominating + Proper Coloring” in Graphs

A. P. Kazemi1

Abstract

Let G be a finite simple graph. A total dominating set in G is asubset of the vertices in G such that each vertex in it is adjacent to atleast one vertex in it. The total domination number of G is the order ofa minimum total dominating set in G. Also a proper coloring of G is afunction from the vertices of the graph to a set of colors such that anytwo adjacent vertices have different colors. The chromatic number of G isthe minimum number of colors needed in a proper coloring of a graph. Inthis talk, we will present a new concept, namely total dominator coloring,which is obtained by mixing the previous two concepts, and we will studyit on several classes of graphs, as well as finding general bounds andcharacterizations. We also show the relation between total dominatorchromatic number with some other numbers in graphs.

References

[1] David F. Anderson and Philip S. Livingston, The zero-divisor graph of a commu-

tative ring, J. Algebra, 217(1999), 434-447.

[2] Thomas G. Lucas, The diameter of a zero-divisor graph, J. Algebra, 301(2006),

174-193.

[1] R. Gera, On dominator colorings in graphs, Graph Theory Notes, N. Y. LII

(2007) 25-30.

[2] T. W. Haynes, S. T. Hedetniemi and P. J. Slater (Eds.), Fundamentals of Domi-

nation in Graphs, Marcel Dekker, Inc. New York, 1998.

[3] M. Chellali and F. Maffray, Dominator Colorings in Some Classes of Graphs,

Graphs and Combinatorics, 28 (2012) 97-107.

[4] M. A. Henning and A. P. Kazemi, k-tuple total domination in graphs, Discrete

Applied Mathematics, 158 (2010) 1006-1011.

1Department of Mathematics and Applications, College of Mathematical Sciences, Uni-versity of Mohaghegh Ardabili, Ardabil, Iran, [email protected]

68



Application of Laurent Series Ring in WirelessCommunication

D. Kiani1 and H. Khodaiemehr∗2

Abstract

We construct full-diversity, arbitrary rate space-time block codes overany a priori specified signal set for a perfect square number of transmitantennas using twisted Laurent series ring.

1 Introduction

The aim of this note is to bring to the attention of a wide mathematicalaudience the recent application of division algebras to wireless communication.The application occurs in the context of communication involving multipletransmit and receive antennas, a context known in engineering as MIMO,short for multiple input, multiple output. Our focus here will be on algebraicaspect of this subject.Interest in MIMO communication began where it was established that MIMOwireless transmission could be used both to decrease the probability of error aswell as to increase the amount of information that can be transmitted. Signaldesign for such situations, i.e., for fading channels with multiple transmit andreceive antennas is called space-time coding (STC) [4]. A class of STC calledspace-time block coding (STBC) has attracted wide attention because of theavailability of low-complexity decoders for them. From [2], an n × l, (n ≤ l)STBC C is a finite set of n× l matrices (codewords) over the complex field C.By the following definition we can describe STBCs for n transmit antennaswith a signal set S and a matrix which avoids exhaustive listing of codewordsof an STBC.

Definition 1. A rate-k/n, n × l linear design over a field F ⊆ C is ann× l matrix with all its entries F-linear combinations of k variables and theircomplex conjugates, which are allowed to take values from the field F. If we

1Department of Mathematics and Computer Science, Amirkabir University of Technology(Tehran Polytechnic), 424, Hafez Ave., Tehran 15914, Iran, [email protected]

2Department of Mathematics and Computer Science, Amirkabir University of Technology(Tehran Polytechnic), 424, Hafez Ave., Tehran 15914, Iran, [email protected]

69

restrict the variables to take values from a finite subset S of F, then we getan n× l STBC C over that finite subset S, for n transmit antennas.

Let C ⊂ Mn(C) be a space-time block code. In order for C to perform well,it should satisfy property (1) below (as remarked before) and as many of theother properties as possible.

(1) It is fully diverse: det(X −X ′) 6= 0 for all matrices X 6= X ′, X,X ′ ∈ C.

(2) It has full rate, which means that the n2 degrees of freedom are used totransmit n2 information symbols. It is high-rate if the rate of the codeis greater than 1.

2 Twisted Laurent Series Ring

In this section, we give a brief introduction to twisted Laurent rings. let L bea commutative field and σ be an automorphism of L. Call K = FixL(σ) =x ∈ L | σ(x) = x the fixed field of σ in L.

Definition 2. Denote by L((T, σ)) the ring of formal Laurent series∑∞

i=R aiTi

in the indeterminate T with coefficients ai ∈ L and (R ∈ Z), with usual ad-dition but skew multiplication such that Ta = σ(a)T , i.e., T ia = σi(a)T i,(a ∈ L).

It is shown in [1] that L((T, σ)) is always a skew field (or a division ring).

Now let L again be a commutative field, then we have the following lemma,[1]:

Lemma 3. Let D = L((T, σ)) be given. If σ has infinite order, then Z(D) =K, hence [D : Z(D)] = ∞; if σ has the finite order n in Aut(L), then Z(D) =K((Tn)), hence [D : Z(D)] = n2 (K = FixL(σ)).

If σ has the finite order n, we want to calculate the dimension |D : Z(D)| in thelatter case, first we observe L is a Galois extension of K and |L : K| = n (seeGalois Theory, in particular Artin’s Lemma [3]). For L/K is obviously cyclicwith generating automorphism σ. Now choose a basis

1, t, t2, t3, ..., tn−1

of L

as aK-space (in the case of separable extensions always this basis exists and wecan see that L = K(t) for a primitive element t ∈ L), then our considerationsshow immediately that

tiT j |0 ≤ i, j < n

is a basis of D as a K((Tn))-space,

hence |D : Z(D)| = n2.

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3 STBCs from Twisted Laurent Series Ring

In the previous section, we gave the definition of twisted laurent rings, and wesaw that D = L((T, σ)) is a division algebra over it’s center. Now we want toconstruct a class of STBCs from these division algebras by embedding theminto the ring of matrices. Consider D = L((T, σ)) and let F = Z(D), then wecan view D as a left F-space, i.e., the action of scalar multiplication is givenby left multiplication. In this section, we use this property and constructarbitrary rate, full-rank STBCs.

Theorem 4. If σ be an automorphism in Galois group of L/K and ord(σ) =n, with D = L((T, σ)) and Tn = x, the set of matrices of the form describedin previous paragraph have the property that the difference of any two suchmatrices is invertible. In addition we have D →Mn2 (K((x)) ).

In representation of the matrixes in Theorem 4, fijs are series in terms ofindeterminate x = Tn, we can consider them in special case to be polynomialsin terms of x. From the preceding theorem it is clear that if K is a subfield ofC and, if we restrict coefficients of fij to some finite subset S of K, we will geta finite set of n2 × n2 matrices and the STBC defined by this set of matriceswill be for n2 transmit antennas, arbitrary-rate, and it will be of full-rankbecause D is a division algebra (we can increase the degree of polynomialshow much we need so the rate of the code can be determined by ourselves).In the preceding example, we obtained a full-rank STBC from a Laurent ringby using theorem 4.

Example 5. Let S be an arbitrary signal set (for example we can considerQAM or PSK signal sets). We want to construct a full-rank STBC for 4transmit antennas. We put L = Q(S)(

√2) and D = L((T, σ)), then by the

notation used t =√2 and if we consider σ : L → L, σ(

√2) = −

√2 and

σ = id for elements which are in Q(S), then n = ord(σ) = 2 and K =FixL(σ) = Q(S) and form Lemma 3, Z(D) = F = K((T 2)) = Q(S)((x))where T is an indeterminate and T 2 = x. It is clear that L/K is Galoisextension, and the basis of D = L((T, σ)) over F is B =

1,√2, T,

√2T

. If

β = f0 + f1√2+ f2T + f3

√2T then, with this notations, the STBC we obtain

will have codewords of the form

Cβ =

f0 2f1 xf2 −2xf3f1 f0 −xf3 xf2f2 2f3 f0 −2f1f3 f2 −f1 f0

; (1)

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where fi , i = 0, 1, 2, 3, is of the form fi =∑∞

k=Rifk,ix

k and fk,i ∈ K = Q(S)for all k ≥ Ri. If we want to have the rate of R bits per channel use, it’s enoughto put fi =

∑R0 fk,ix

k where fk,i now come from arbitrary finite subset of thefield Q(S) ⊆ C.Now it’s enough to replace indeterminate x with appropriate element in Cto have a STBC with rate R for 4 transmit antennas over signal set S. it’senough to replace x in (1) by α = ω83 and the STBC we obtain will be fulldiversity.Now we present second candidate of indeterminate x which is a transcenden-tal number. The following theorem, namely Lindemann-Weierstrass theorem,suggests a method to find algebraically independent transcendental numbers.

Definition 6. Let F be an field extension of K and S be a subset of F.We call S is algebraically dependent over K, if for a natural number n, thereexist a non-zero polynomial f ∈ K [x1, . . . , xn] such that for disjoint elementss1, . . . , sn ∈ S, f(s1, . . . , sn) = 0. S is algebraically independent over K, if it’snot algebraically dependent.

Theorem 7. If u1, u2, . . . , un are algebraic numbers that are linearly inde-pendent over Q, then the exponentials eu1 , eu2 , . . . , eun are algebraically inde-pendent over the field of algebraic numbers.

By using Theorem 7 we can put x = ej√2 (or any transcendental element over

Q) in (1).

References

[1] P. K. Draxl, “Skew Fields,” Cambridge Univ. Press, Cambridge, 1983.

[2] J. C. Guey, M. P. Fitz, M. R. Bell, and W. Y. Kuo, Signal design for transmitterdiversity wireless communication systems over Rayleigh fading channels, in Proc.IEEE Vehicular Technology Conf., Atlanta, GA, Apr. 1996, pp. 136140, Also,IEEE Trans. Commun., 47 , no. 4, Apr. (1999), pp. 527–537.

[3] T. W. Hungerford, “Algebra,” 3rd edition, Springer-Verlag, Washington, 1980.

[4] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-time codes for highdata rate wireless communication: Performance criterion and code construction,IEEE Trans. Inf. Theory, 44, no. 2, Mar. (1998), pp. 744–765.

72



Completely Irreducible Submodules and SomeCharacterizations of Distributive Modules

A. Khojali1

Abstract

A submodule of an R-module, R a ring, is called completely irre-ducible if it is not the intersection of any set of its over families. Weinvestigate the structure of completely irreducible submodules in an R-module without finiteness conditions. It is well known that a submoduleis an intersection of an over family of completely irreducible submodules.Submodules that admit a representation as an irredundant intersectionof an over family, of completely irreducible submodules, are character-ized in several ways. By using these observations, we prove that over aNoetherian ring R, an R-module is Artinian iff every submodule is an in-tersection of a finite over family of completely irreducible submodules iffits zero submodule is an intersection of a finite over family of completelyirreducible submodules. Also, we observe several characterizations ofdistributive modules. In particular, we prove that an R-module M isdistributive iff its completely irreducible submodules is the set

mRx(m) : x ∈M, m ∈Max(R) and m ∈ Supp(Rx)

Definition 1. Let M be an R-module. A proper submodule of M , is calledcompletely irreducible, if it is not the intersection of any over family.

Theorem 2. Let M be an R-module. If N is a proper submodule of M , thenthe following conditions are equivalent:

(i) M/N is a subdirectly irreducible R-module;

(ii) N is a completely irreducible submodule;

(iii) The socle of M/N is a simple essential submodule;

(iv) N is irreducible and M/N contains a simple submodule;

(v) N is irreducible, and properly contained in N :M m for some maximalideal m of R;

1Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ard-abil, Iran, [email protected], [email protected]

73

(vi) N is irreducible with adjoint prime a maximal ideal m of R, and m =N :R x for some x ∈M \N ;

(vii) N = N(m) for some maximal ideal m of R, and Nm is a completelyirreducible submodule of Mm.

Proposition 3. Let K ⊆ N be submodules of an R-module M . Then, thefollowing statements are equivalent:

(i) N is a completely irreducible divisor of K;

(ii) N/K is not an essential submodule of M/K;

(iii) Soc(N/K) is properly contained in N∗/K.

Definition 4. An intersection⋂

i∈I Ki of submodules of an R-module M issaid to be irredundant if each component of the intersection is relevant. In thesense that omitting some Ki makes the intersection

⋂i∈I Ki larger.

Definition 5. Let Kii∈I is a family of R-modules. By an interdirect prod-uct of this family we mean a submodule of the direct product

∏i∈I Ki, that

contains the direct sum⊕

i∈I Ki.

Lemma 6. Let M be an R-module. Let N =⋂

i∈I Ki, be an irredundantrepresentation of the submoduleN , whereKi (i ∈ I) are completely irreducible

submodules. For each j ∈ I let Kj :=⋂

i∈I\jKi, then

(i) There are elements mi(i ∈ I) such that K∗i = Ki +Rmi.

(ii) For each i ∈ I, Ki/N has an essential simple socle generated by mi +N.

(iii) In the representation N =⋂

i∈I Ki, no Ki can be replaced by a largersubmodule, and still have the intersection be equal to N.

(iv) With mi as in part (i), let Mi := (Rmi + N)/N. Then Mi is a simpleR-module, and Soc(M/N) is an interdirect product of Mii∈I .

Theorem 7. Let M be an R-module, and N a proper submodule of M. Thefollowing statements are equivalent:

(i) N is an irredundant intersection of an over family of completely irre-ducible submodules;

74

(ii) M/N is an irredundant subdirect product of subdirectly irreducible R-modules;

(iii) The injective hull E(M/N) of M/N is an interdirect product of theinjective hulls of a set of simple R-modules.

Corollary 8. Let R be a Noetherian ring. The following statements areequivalent:

(i) Every proper submodule of M is an intersection of a finite over familyof completely irreducible submodules;

(ii) The zero submodule is an intersection of a finite over family of completelyirreducible submodules;

(iii) Soc(M) is a finitely generated essential submodule of M ;

(iv) M is Artinian.

Theorem 9. Let the submodule N of M is an irredundant intersection (pos-sibly infinite) of an over family of completely irreducible submodules. Thenthe socle ofM/N is essential. The converse holds, if the statement of Theoremis true for every submodule.

Theorem 10. Let M be an R-module. Then, the following statements areequivalent:

(i) M is distributive;

(ii) Mm is a uniserial Rm-module;

(iii) For any pair X ⊆ Y of submodules of M , Y finitely generated, thereexists an ideal c such that X = cY.

Theorem 11. Let M be an R-module. Then, the following statements areequivalent:

(i) M is distributive;

(ii) (X+Y : Z) = (X : Z)+ (Y : Z) for arbitrary submodules X,Y and anyfinitely generated submodule Z;

75

(iii) (X : Y ∩ Z) = (X : Y ) + (X : Z) for any finitely generated submodulesY, Z and arbitrary submodule X.

Theorem 12. Let R be a semi-local ring. If M is a distributive R-module,then it would be a Bezot module.

Theorem 13. Let (R,m) be a quasi-local ring. For an R-module M thefollowing statements are equivalent:

(i) M is uniserial;

(ii) If N is a completely irreducible submodules of M , then there exists amaximal ideal m and a nonzero element x ∈M such that N = mx;

(iii) For each nonzero element x ∈M,mx is an irreducible submodule of M .

Theorem 14. Let M be a module over a ring R. Then, M is distributive iffits completely irreducible submodules is the set

mRx(m) : m ∈Max(R), x ∈M and m ∈ Supp(Rx).

References

[1] A. Walendziak, Meet decomposition in complete lattices, Per. Math. Hung. 21(3)

(1990), 219-222.

[2] C. Jensen, Arithmetical rings, Acta. Math. Acad. Sci. Hung. 17 (1966), 115-123.

[3] J. Dauns, Primal Modules, Comm. Algebra, 25, 8 (1997), 2409-2435.

[4] L. Fuchs, On primal ideals, Proc. Amer. Math. Soc. 1 (1950), 1-6.

[5] L. Fuchs, W. Heinzer, B. Olberding, Commutative ideal theory without finiteness

conditions: Completely irreducible ideals: Trans. Amer. Math. Soc. 358 (2006),

3113-3131.

76



Proof of Cantor’s Hypothesis

J. Kurdics1

Abstract

In this talk shall attempt to prove Cantors Hypothesis, a special caseof which is known as the Continuum Hypothesis, namely that the car-dinality of the power set of an infinite set is the consecutive cardinality.An ordered field of cardinality ℵα with interval topology of weight ℵα isconstructed, where ℵα is an uncountable isolated cardinal.

We list here some results:

Theorem 1. Cantor’s Hypothesis: Cardinality of the power set of an infiniteset is the consecutive cardinality.

Let ω and Ω be the first countably infinite and the first uncountable ordinalof cardinalities ℵ0 and ℵ1, respectively. The well-known special case:

Theorem 2. Continuum Hypothesis: Cardinality of the power set of ω is thecardinality ℵ1.

Construct a number system ”of base ω” for countable ordinals. For any count-able ordinal α, let τ0 = 1, and

τα = supτβ ∗ j |β < α, j ∈ ω.

Theorem 3. Any nonzero countable ordinal γ can be written uniquely asγ =

⊕nj=1 ταj ∗ kj with α1 > α2 > · · · > αn, the kj ∈ ω are all nonzero.

Let ZΩ be the free left module over the ring of integers Z with well-orderedbasis τµ |µ ∈ Ω. The set Ω of countable ordinals has a natural embeddingf : Ω → ZΩ, 0 7→ 0,

⊕ni=1 τµi ∗ ki 7→

∑ni=1 kiτµi . The well-order of the basis

induces an order on the module ZΩ. Let the positivity domain ZΩ+ be the set

of all elements with positive leading coefficients k1.

Theorem 4. The pair (ZΩ,+) becomes an ordered free abelian group ofcardinality ℵ1.

1Nyıregyhaza College, Hungary, [email protected]

77

Define multiplication of basis elements of the module by the rule τµτν =τf−1(f(µ)+f(ν)), where f is the natural embedding. Extend the multiplicationby distributivity.

Theorem 5. (ZΩ,+, ·) becomes an ordered integral domain.Let QΩ be the quotient field of the integral domain with positivity domain Q+

Ω

the set of all fractions with numerator and denominator both either positiveor negative.

Theorem 6. QΩ is a (linearly) ordered field of cardinality ℵ1, endowed withthe interval topology.

Theorem 7. A strictly increasing sequence uii∈ω+ in the unit interval [0, 1]of the field QΩ has an upper bound h so that h− ε is not an upper bound foran arbitrarily small ε ∈ Q+

Ω , called an ε-least upper bound.A nondegenerate interval of the field QΩ contains uncountably many elements.

Theorem 8. In the field QΩ the intersection of a strictly decreasing sequenceof closed intervals within the unit interval contains a nondegenerate closedinterval.Apply the construction of Cantor’s triadic set in the field QΩ. It followsthat QΩ contains a subset of continuum cardinality. Consequently ℵ1 is thecardinality of the power set of a set of cardinality ℵ0.Proof of the general Hypothesis is completely analogous.

78



Invariant Finsler Metrics on Lie Groups

D. Latifi1

Abstract

In this paper we study the geometry of Lie groups with left invariantFinsler metrics. We give some algebraic descriptions of these metrics.

1 Introduction

The study of invariant structures on Lie groups and homogeneous spaces isan important problem in differential geometry and algebra of Lie groups. Liegroups are, in a sense, the nicest examples of manifolds and are good spaceson which to test conjectures [3]. Therefore it is important to study invariantFinsler metrics. Among the invariant metrics the bi-invariant ones are thesimplest kind. They have nice and simple geometric properties, but still forma large enough class to be of interest. In [2] we have studied the bi-invariantRansers metrics on Lie groups.

The purpose of this paper is to study the geometric and algebraic proper-ties of left invariant Finsler metrics on Lie groups. We give some algebraicdescriptions of these metrics. We show that for a compact Lie group withbi-invariant absolutely homogeneous Finsler metric F if the flag curvature ofF is everywhere nonzero then F is Riemannian .

2 Left invariant Finsler metrics on Lie groups

Let M be an n−dimensional smooth manifold without boundary and TMdenote its tangent bundle. A Finsler structure on M is a map F : TM −→[0,∞) which has the following properties [1]:

1. F is smooth on TM := TM\0.

2. F (x, λy) = λF (x, y), for any x ∈M,y ∈ TxM and λ > 0.

1Department of Mathematics and Application, University of Mohaghegh Ardabili, Ard-abil, Iran, [email protected]

79

3. F 2 is strongly convex, i.e.,

gij(x, y) :=1

2

∂2F 2

∂yi∂yj(x, y)

is positive definite for all (x, y) ∈ TM .

Let G be a connected Lie group with Lie algebra g = TeG. A Finsler functionF : TG −→ R+ on Lie group G is called left-invariant if

F (x, y) = F (La(x), (La)∗x(y))

for all a, x ∈ G and y ∈ TxG. Similarly, a Finsler metric is right-invariant ifeach Ra : G −→ G is an isometry.

A Finsler metric on G that is both left-invariant and right-invariant is calledbi-invariant.

Theorem 1. Let G be a connected Lie group furnished with a left-invariantFinsler metric F . Then the following are equivalent,

1. F is right-invariant, hence bi-invariant.

2. F is Ad(G)−invariant.

3. gY ([X,U ], V ) + gY (U, [X,V ]) + 2CY ([X,Y ], U, V ) = 0, ∀ Y ∈ g −0, X, U, V ∈ g, where Cy is the Cartan tensor of F at Y .

If the Finsler structure F is absolutely homogeneous, then one also has.

4. The inversion map g −→ g−1 is an isometry of G.

Proof: The equivalence of the first two assertion is routine, and we omit thedetails.(2) =⇒ (3). Since F is Ad(G)−invariant, we have

F 2(Ad(exp(tX))(Y + rU + sV )) = F 2(Y + rU + sV ).

Thus

gY (U, V ) =1

2

∂2

∂r∂sF 2(Ad(exp(tX))(Y + rU + sV ))|r=s=0.

Taking derivative with respect to the t at t = 0, we get

0 = gY ([X,U ], V ) + gY (U, [X,V ]) + 2CY ([X,Y ], U, V ).

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(3) =⇒ (2). For any Y 6= 0, U, V,X ∈ g, consider the function

ψ(t) = gAd(exp(tX))Y (Ad(exp(tX))U,Ad(exp(tX))V ) .

Then taking the derivative with respect to t, we haveψ′(t) = 0. Therefore the function ψ is constant, hence

gY (U, V ) = gAd(exp(tX))Y (Ad(exp(tX))U,Ad(exp(tX))V ) ∀t ∈ R.

Since G is connected, it is generated by element of the form exp(tX), X ∈g, t ∈ R. Thus

F (Ad(g)Y ) = F (Y ), ∀g ∈ G, Y ∈ g.

If F is absolutely homogeneous, one can check quite easily that property (4)is equivalent to property (1).2

Theorem 2. Let G be a compact Lie group with a bi-invariant absolutelyhomogeneous Finsler metric F . If the flag curvature of (G,F ) is everywherenonzero, then F is Riemannian.

References

[1] D. Bao, S. S. Chern and Shen , An Introduction to Riemann-Finsler geometry,

Springer-Verlag, New-York, (2000),

[2] D. Latifi, Bi-invariant Randers metrics on Lie groups, Publ. Math. Debrecen,

76(2010), 219-226.

[3] J. Milnor, curvature of left invariant metrics on Lie groups, Advances in Math.,

21(1976), 293-329.

81



Simple Power Graphs of Semigroups

S. Mohamadikhah∗1 and N. Vaez Moosavi2

Abstract

Let S be a semigroup. The power graph of S, denoted by g(S), is anundirected graph with S as the set of its vertices and

∀a, ba6=b

∈ S : a− b ⇔ am = b or bm = a for some m ∈ N.

In this paper, we prove that g(S) (where S is finite ) is connected ifand only if S contains a single idempotent. Also we consider the graphg(Zn) for the multiplicative semigroup Zn of modulo n, together with anapplication example.

References

[1] J. Bosak, The graphs of semigroups, In. Theory of Graphs and Application, Aca-

demic Press, New York (1964) 119-125.

[2] F. Budden, Cayley graphs for some well-known groups, Math. Gaz. 69 (1985)

271-278. [3] B. Zelinka, Intersection graphs of finite Abelian groups, Czech Math. J.

25(100) (1975) 171-174.

1Khoy Payam Noor University, [email protected] Payam Noor University, [email protected]

82



On Graded Classical Prime Submodules

S. Motmaen1

Abstract

Let R be a G-graded commutative ring with identity and let M be agraded R-module. A proper graded submodule N of M is called gradedclassical prime if for every a, b ∈ h(R), m ∈ h(M), whenever abm ∈ N ,then either am ∈ N or bm ∈ N . The spectrum of graded classical primesubmodules of M is denoted by Cl.Specg(M).In this talk we study somebasic properties of these classes of graded classical prime submodule ofgraded R-module M .

1 Introduction

A grading on a ring and its modules usually aids computations by allowingone to focus on the homogeneous elements, which are presumably simpler ormore controllable than random elements. However, for this to work one needsto know that the constructions being studied are graded. One approach tothis issue is to redefine the constructions entirely in terms of the category ofgraded modules and thus avoid any consideration of non-graded modules ornon-homogeneous elements; Sharp gives such a treatment of attached primesin [9]. Unfortunately, while such an approach helps to understand the gradedmodules themselves, it will only help to understand the original constructionif the graded version of the concept happens to coincide with the original one.Therefore, notably, the study of graded modules is very important.

Our main purpose is to study some new classes of graded submodules of gradedmodules . Therefore these results will be used in order to obtain the main aimsof this talk.

2 Graded classical prime submodules

A proper graded submodule N ofM is called graded classical prime if for everya, b ∈ h(R), m ∈ h(M), whenever abm ∈ N , then either am ∈ N or bm ∈ N .

1Young Researchers Club, Ardabil branch Islamic Azad University, Iran,[email protected]

83

Let N be a graded classical prime submodule of M . Then, it is easy to seethat Ng is a classical prime submodule of the Re-module Mg for every g ∈ G.It is evident that every graded prime submodule is graded classical prime.However the next example shows that a graded classical prime submodule isnot necessarily graded prime.

Example 1. Assume that R is a graded integral domain and P is a non-zerograded prime ideal of R. In this case the ideal Q := P ⊕0 is a graded classicalprime submodule of the graded R-module R⊕R while it is not graded prime.This example shows also that a graded classical prime submodule need not beclassical prime.

The following lemma is obvious.

Lemma 2. Let N be a proper graded submodule of M . Then N is a gradedclassical prime submodule if and only if for each x ∈ h(M) \N , (N :R x) is agraded prime ideal of R.

Proposition 3. (1) Let N be proper graded submodule of M . Then N is agraded prime submodule of M if and only if N is graded primary and gradedclassical prime.

(2) Assume that N and K are graded submodule of M with K ⊆ N . ThenN is a graded classical prime submodule of M if and only if N/K is a gradedclassical prime submodule of the graded R-module M/K.

In [10], Specg(M) has endowed with quasi-Zariski topology. For each gradedsubmodule N of M , let V g

∗ (N) = P ∈ Specg(M)|N ⊆ P. In this case,the set ζg∗ (M) = V g

∗ (N)|N is a graded submodule of M contains the emptyset and Specg(M), and it is closed under arbitrary intersections, but it is notnecessarily closed under finite unions. The graded R-module M is said to bea g-Top module if ζg∗ (M) is closed under finite unions. In this case ζg∗ (M)satisfies the axioms for the closed sets of a unique topology τ g∗ on Specg(M).The topology τ g∗ (M) on Specg(M) is called the quasi-Zariski topology. In theremainder of this section we use a similar method to define a topology onCl.Specg(M). To this end, For each graded submodule N of M , set

Vg∗(N) = P ∈ Cl.Specg(M)|N ⊆ P.

Proposition 4. Let M be a graded R-module. Then

(1) For each subset E ⊆ h(M), Vg∗(E) = Vg

∗(N) = V g∗ (Gr

clM (N)), where N is

the graded submodule of M generated by E.

(2) Vg∗(0) = Cl.Specg(M), and Vg

∗(M) = ∅.

84

(3) If Nλλ∈Λ is a family of graded submodules of M , then⋂

λ∈ΛVg∗(Nλ) =

Vg∗(∑

λ∈ΛNλ).

(4) For every pair N and K of graded submodules of M , we have Vg∗(N) ∪

Vg∗(K) ⊆ Vg

∗(N ∩K).

Now if we set

ηg∗(M) = Vg∗(N)|N is a graded submodule of M

then ηg∗(M) contains the empty set and Cl.Specg(M). Moreover ηg∗(M) isclosed under arbitrary intersections, but it is not necessarily closed underfinite unions.

Definition 5. Let M be a graded R-module.

(1) We shall say that M is a g-Cl.Top module if ηg∗(M) is closed under finiteunions, i.e. for any graded submodules N and L of M there exists a gradedsubmodule K of M such that Vg

∗(N) ∪ Vg∗(L) = Vg

∗(K).

(2) A graded classical prime submodule N of M will be called graded classicalextraordinary, or g-Cl.extraordinary for short, if wheneverK and L are gradedclassical semiprime submodules of M with K ∩L ⊆ N then K ⊆ N or L ⊆ N .

The next result is a useful tool for characterizing g-Cl.Top modules.

Theorem 6. Let M be a graded R-module. Then, the following statementsare equivalent:

(i) M is a g-Cl.Top module.

(ii) Every graded classical prime submodule of M is g-Cl.extraordinary.

(iii) Vg∗(N)∪Vg

∗(L) = Vg∗(N∩L) for any graded classical semiprime submodules

N and L of M .

Corollary 7. Every g-Cl.Top module is a g-Top module.

Theorem 8. Let M be a g-Cl.Top R-module. Then,

(1) For every graded submodule K of M , the R-module M/K is a g-Cl.Topmodule.

(2) The graded RP -module MP is a g-Cl.Top module for every graded primeideal P of R.

(3) If GrclM (N) = N for every graded submodule N of M , then M is a gradeddistributive module.

For every subset S of h(M), define

XS = X − Vg∗(S)

85

In particular, if S = f, we denote XS be Xf .

Proposition 9. The set Xf |f ∈ h(M) is a basis for the quasi-Zariskitopology on X.A topological space X is said to be irreducible if X 6= ∅ and if every pair ofnon-void open sets in X intersect. Let X be a topological space. A subsetA ⊆ X is said to be dense in X if and only if A ∩ G 6= ∅ for every non-voidopen subset G ⊆ X. Therefore X is irreducible if and only if every non-voidopen subset of X is dense.

Lemma 10. Let M be a graded R-module. Then, N := GrclM (0) is a gradedclassical prime submodule of M if and only if Cl.Specg(M) is irreducible.

References

[1] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Longman

Higher Education, New York 1969.

[2] M. Behboodi and H. Koohi, Weakly prime modules, Vietnam J. Math. 32(2),

185-195 (2004).

[3] S. Ebrahimi Atani, On graded prime submodules, Chiang Mai J. Sci., 33(1), 3–7

(2006).

[4] S. Ebrahimi Atani and F. Farzalipour, On weakly prime submodules, Tamkang

Journal of Mathematics, 38(3), 247–252 (2007).

[5] S. Ebrahimi Atani and F. Farzalipour, On graded multiplication modules, Chiang-

Mai Journal of Science, To appear.

[6] C. P. Lu, The Zariski topology on the prime spectrum of a module, Houston J.

Math., 25(3), 417–425 (1999).

[7] R. L. McCasland, M. E. Moore and P. F. Smith, On the spectrum of a module over

a commutative ring, Comm. Algebra, 25 (1), 79–103 (1997).

[8] K. H. Oral, U. Tekir, and A. G. Agargun, On graded prime and primary submod-

ules, Turk. J. Math., 25(3), 417–425 (1999).

[9] R. Y. Sharp, Asymptotic behavior of certain sets of attached prime ideals, J. London

Math. Soc., 212-218, (1986).

[10] A. Yousefian Darani, Topologies on Specg(M), Buletinul Academiei de Stiinte a

Republicii Moldova Matematica, To appear.

86



A Survey of k-tuple Total Domination in Graphs

A. P. Kazemi1 and B. Pahlavsay∗2

Abstract

A set S of vertices in a graph G is a k-tuple total dominating set ofG if every vertex of G is adjacent to least k vertices in S. The minimumcardinality of a k-tuple total dominating set of G is the k-tuple totaldomination number of G and denoted by γ×k,t(G). For a graph to havea k-tuple total dominating set, its minimum degree is at least k. Inthis paper, we offer a survey of selected recent results on k-tuple totaldomination in graphs, which are appeared in the following references.

References

[1] A. P. Kazemi, k-tuple total domination and Mycieleskian graphs, Transactions On

Combinatorics, 1 (2012) 7-13.

[2] A. P. Kazemi, k-tuple total domination in complementary prisms, ISRN Discrete

Mathematics , Article ID 681274 (2011) 1-13.

[3] A. P. Kazemi, k-tuple total domination in inflated graphs, FILOMAT, to appear

(2012), .

[4] A. P. Kazemi and B. Pahlavsay, k-tuple total domination in Supergeneralized Pe-

tersen graphs, Communications in Mathematics and Applications, 2 No.1 (2011)

2130.

[5] A. P. Kazemi and B. Pahlavsay, double total domination in Harary graphs, Open

Journal of Discrete Mathematics, accepted (2012).

[6] A. P. Kazemi and B. Pahlavsay, k-tuple total domination number of cartesian

product graphs, Miskolc Mathematical Notes, submited (2012).

[7] D. Pradhan, Algorithmic aspects of k-tuple total domination in graphs, Information

Processing Letters, 112 (2012) 816822.

1Department of Mathematics, University of Mohaghegh Ardabili, P.O.Box 5619911367,Ardabil, Iran, [email protected]

2Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran,[email protected]

87

[8] M. A. Henning and A. P. Kazemi, total domination in inflated graphs, Discrete


[9] M. A. Henning and A. P. Kazemi, k-tuple total domination in graphs, Discrete


[10] M. A. Henning and A. P. Kazemi, k-tuple total domination number of cross

products graphs, J Comb Optim , (2011) DOI 10.1007/S10878-011-9389.

88



On Minimax and Artinian Modules

N. Rafi’ei∗1 and J. Azami2

Abstract

Let R be a commutative Noetherian ring and let M be a unitary R-module. In this paper we investigate some new properties about reduced,coatomic and complemented R-modules.

Definition 1. An R-module M is called coatomic, if every proper submoduleof M is contained in a maximal submodule of M .

Definition 2. An R-module M is called radical, if it has no maximal sub-module. By P (M) we denote the sum of the radical submodule of M . P (M)is the largest radical submodule of M . If P (M) = 0, M is called reduced.

Definition 3. AnR-moduleM is called complemented if, for every submoduleNof M , the set K ⊆M : N +K =M has a minimal element.

Definition 4. A proper submodule N of An R-module M is called small, iffor every submodule K of M , with K +N =M then K =M .

Theorem 5. Let R be a Noetherian ring, and M be an R-module.(i)If M is a minimax and Ass(M) consist of maximal ideal of R, then M isArtinian.

Theorem 6. Let R be a Noetherian ring, and let M be a complementedR-module. Then M/P (M) is coatomoc.

Theorem 7. Let M be a non-Artinian R-module and I be an ideal of Rsuch that In = 0 for some positive integer n. Then M/IM is not ArtinianR-module.



89

Example 8. Let (R,m) be a Noetherian local ring and E := E(R/m) theinjective envelop of R/m. Then the R-module ⊕n(0 :E mn) is reduced andfaithful.

Theorem 9. LetM be an R-module and I be an ideal of R such that InM = 0for some positive integer n. Then IM is a small submodule of M .

Theorem 10. Let R be a Noetherian ring, and let N ⊆M be an R-modulessuch that M/N is Artinian (Noetherian) R-module. Then for each ideal Iof R and for each positive integer n, the R−module InM/InN is Artinian(Noetherian) respectively.

References

[1] E. Matlis, Injective modules over Noetherian rings, Pacific J. Math, 8(1958), 511-

528.

[2] E. Matlis, 1-Dimensional Cohen-Macaulay Rings, Lecture Notes in Mathematics,

Springer, Berlin, 327(1973).

[3] P. Rudlof, On the sttructure of couniform and complemented modules, J. pure

Appl. AlgebraMath, 74(1)(1991), 281305.

90



The Expansion Functor on a Bipartite Graph

R. Rahmati-Asghar1

Abstract

For α ∈ Nn, suppose that Gα be a simple graph obtained from ap-plying the expansion functor on the simple graph G with respect to α.A class of simplicial complexes, called Cohen-Macaulay of degree k (orCM(k) for short) is defined and it is shown that a bipartite graph G isCohen-Macaulay if and only if Gα is CM(k) where α = k.1. Also, all bi-partite graphs which are CM(k) are characterized. Our result improvesa result due to Herzog and Hibi and also a result due to Estrada andVillarreal.

1 Introduction

Let G be a simple graph with finite vertex set V = x1, . . . , xn. We denoteby V (G) and E(G) the set of vertices and edges of G, respectively. Let S =K[x1, . . . , xn] be a polynomial ring over a fieldK. The edge ideal of G, denotedby I = I(G), is the ideal of G generated by all squarefree monomials xe =∏xi∈e

xi such that e ∈ E(G).

The duplication of a vertex xi of a simple graph G means extending its vertexset V (G) by a new vertex x′i and replacing E(G) by

E(G) ∪ (e\xi) ∪ x′i : e ∈ E(G).

Let α = (k1, . . . , kn) ∈ Nn. The simple graph Gα is obtained from G bysuccessively duplicating ki − 1 times every vertex xi if ki ≥ 1 (c.f. [1], [5] and[6]).

In this paper, first we introduce a class of simplicial complexes, called Cohen-Macaulay simplicial complexes of degree k (CM(k) simplicial complexes forshort) which generalizes the notion of Cohen-Macaulayness for simplicial com-plexes. We say that a simple graph is CM(k) if its independence complex isCM(k).

1School of Mathematics, Statistics and Computer Science, College of Science, Universityof Tehran, Tehran, Iran, [email protected]

91

Next we introduce a class of monomial ideals with k-resolution. It is shownthat G is CM(k) if and only if the vertex cover ideal of Gα has a k-resolutionwhich 1 ≤ k ≤ d. As a main result, we characterize all bipartite graphs whichare CM(k). Our result generalizes Theorem 2.9 of [3] and also Theorem 3.4of [4]. It is also concluded that for a positive integer k, an unmixed bipartitegraph G is (shellable) Cohen-Macaulay if and only if Gα is (k-shellable) CM(k)where α = k.1.

2 Cohen-Maulayness of degree k

First, we recall some definitions related to simplicial complexes. Given asimplicial complex ∆ on [n], the Stanley-Reisner ideal of ∆ in S is a monomialideal generated by monomials xF which F 6∈ ∆. The Stanley-Reisner ring of∆ is defined K[∆] := S/I∆.

The link of a face F in ∆ is defined by

link∆(F ) = G ∈ ∆ : F ∩G = ∅, F ∪G ∈ ∆.

Moreover, the Alexander dual of ∆ is defined as ∆∨ = F ∈ ∆ : [n]\F 6∈ ∆.For the subset W of the vertex set of ∆, the restriction of ∆ on W is definedas ∆W = F ∈ ∆ : F ⊆ W. We say that a simplicial complex is pure if allfacets have the same cardinality.

Definition 1. Let ∆ be a simplicial complex of dimension d−1 and 1 ≤ k ≤ dan integer. We say that ∆ is Cohen-Macaulay of degree k (CM(k) for short)if, for all faces F of ∆, Hi−1(link∆F ;K) = 0 unless ik = dim(link∆F ) + 1.

Definition 2. An Nn-graded module M generated in degree b ∈ Nn with|b| = q for some fixed q ∈ N, has a k-resolution if for all i ≥ 0 the minimalith syzygies of M lie in degrees b ∈ Nn with |b| = q + ik. Equivalently,βi,j(M) = 0 for each j 6= q + ik whenever 0 ≤ i ≤ proj.dim(M).

Theorem 3. Let ∆ be a pure simplicial complex of dimension d − 1 and aninteger k with 1 ≤ k ≤ d. Then the following conditions are equivalent:

(a) ∆ is CM(k);

(b) I∆∨ has a k-resolution.

92

3 Main results

For a simple graph G, the independence complex of G is denoted by ∆G andF is a face of ∆G if and only if there is no edge of G joining any two verticesof F . The complement of an independence set is called a vertex set and themonomial ideal generated by all xF which F is a minimal vertex cover of Gis called the vertex cover ideal of G. It is easy to see that I∆G

= I(G) andthe vertex cover ideal of G is equal to I∆∨

G. If ∆G is pure, we say that G is

unmixed.

Let G be a bipartite graph with the vertex partition V ∪ V ′ and the edge setE(G). The following results were proved by the authors in [3, 4]:

• ([3, Theorem 2.9]) G is Cohen-Macaulay if and only if it is pure shellable.

• ([4, Theorem 3.4]) G is Cohen-Macaulay if and only if |V | = |V ′| andthe vertices V = x1, . . . , xn and V ′ = y1, . . . , yn can be labelled suchthat:

(i) xiyi ∈ E(G) for i = 1, . . . , n;

(ii) if xiyj ∈ E(G) then i ≤ j;

(iii) if xiyj , xjyk ∈ E(G) then xiyk ∈ E(G).

Theorem 4. Let G be a bipartite graph without isolated vertices and V ∪V ′

be a vertex partition for G. Let V = x1, . . . , xm and V ′ = y1, . . . , yn.

(a) If k ≥ m and k ≥ n then the following conditions are equivalent:

(i) G is unmixed and CM(k);

(ii) m = n = k and G ∼= Kk,k;

(iii) G is unmixed and k-shellable.

(b) If k < m or k < n then the following conditions are equivalent:

(i) G is CM(k);

(ii) m = n, k|n and the elements of V and V ′ can be labelled such that:

(1) xiyi ∈ E(G) for all i = 1, . . . , n;

(2) if xiyj , xjyl ∈ E(G), then xiyl ∈ E(G);

(3) if xiyj ∈ E(G) then either i ≤ j or there is some l ∈ 0, 1, . . . , n/k−1 such that n− (l + 1)k + 1 ≤ j < i ≤ n− lk;

93

(4) for all l ∈ 0, 1, . . . , n/k − 1, the induced subgraph on the ver-tices xi and yj with n− (l + 1)k + 1 ≤ i, j ≤ n− lk is completebipartite.

(iii) G is unmixed and k-shellable.

Theorem 5. Let G be an unmixed bipartite graph. Let k be a positive integerand α = k.1. Then the following conditions are equivalent:

(i) G is Cohen-Macaulay;

(ii) Gα is CM(k);

(iii) G is shellable;

(iv) Gα is k-shellable.

Corollary 6. Let G be an unmixed bipartite graph. Let k be a positiveinteger and α = k.1. Then the following conditions are equivalent:

(i) the vertex cover ideal of G has linear resolution;

(ii) the vertex cover ideal of Gα has k-resolution;

(iii) the vertex cover ideal of G has linear quotients;

(iv) the vertex cover ideal of Gα has k-quotients;

References

[1] S. Bayati, J. Herzog, Expansions of monomial ideals and multigraded mod-ules, arXiv: 1205.3599v1.

[2] J.A. Eagon, V. Reiner: Resolutions of Stanley-Reisner rings and Alexanderduality. J. of Pure and Appl. Algebra 130, 265-275 (1998).

[3] M. Estrada, R.H. Villarreal, Cohen-Macaulay bipartite graphs, Arch. Math.68, 124-128 (1997).

[4] J. Herzog, T. Hibi, Distributive lattices, bipartite graphs and Alexanderduality, J. Algebraic Combin., 22, 289-302 (2005).

[5] J. Martınez-Bernal, S. Morey, R. H. Villarreal, Associated primes of powersof edge ideals, arXiv:1103.0992v3

94

[6] A. Schrijver, Combinatorial Optimization, Algorithms and Combinatorics24, Springer-Verlag, Berlin, 2003.

95



Invariant Randers Metrics on HomogeneousRiemannian Manifold and Lie Group

Z. Ranjbar∗1 and D. Latifi2

Abstract

This paper studies Randers metrics on homogeneous Riemannianmanifolds. we first give a complete description of the invariant Ran-ders metrics on a homogeneous Riemannian manifold as well as theflag curvature. this result provide a convenient method to constructglobally defined Berwald space which is neither Riemannian nor locallyMinkowskian,then we study the geometry of Lie groups with bi-invariantRanders metric,Finally a necessary and sufficient condition that left in-variant Randers metrics on Lie groups are bi-invariant is given.

1 Introduction

Randers spaces were first introduced by Randers in 1941,when he studiedthe metric problems in 4-space of general relativity.they also accur naturallyin many other physical applications,espacially in electron optics.in geometrythey also provide a rich source of explict examples of y-global Berwald spaces,the constraction of Randers space is not an easy task.this paper provides aconvenient method to construct Randers metrics on homogeneous Riemannianmanifold,as a special case we consider the right and even bi-invariant Randersmetric on Lie groups ([4], [2]).

2 Global expression of Randers metric

Definition 1. Let M be a n-dimensional manifold.A Randers metric on Mconsists of a Riemannian metric a = aijdx

i ⊗ dxj on M and a 1-formb = bi(x)dx

i. By a and b we define a function F on TM

F (x, y) = α(x, y) + β(x, y) x ∈M y ∈ TxM


2Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ard-abil, Iran,

96

where

α(x, y) =√aijyiyj β(x, y) = bi(x)y

i

F is a Finsler structure if and only if

‖b‖ :=√bibi < 1

where bi = aijbj and aij is the inverse of the matrix (aij).

Lemma 2. Let F be a Randers metric on M defined by the Riemannianmetric a and the vector field b].Then (M,F) is of Berwald type if and only ifb] is parallel with respect to a.

Proof. Theorem (11.5.1) of [3] page 301.

2.1 Invariant Randers metric on homogeneous manifols

Let GH be a reductive homogeneous manifold,Let g=LieG, h=Lie H. Fix a

decomposition of g

g = h+m

when m is subspace of g satisfies

Ad(h)m ⊂ m ∀h ∈ H

To constract invariant Randers metric on GH we first need to find invariant

vector fields on GH .

Theorem 3. Let GH be a reductive homogeneous manifold. There exists a

bijection between the set of invariant vector fields on GH and the subspace

V = X ∈ m|Ad(h)(X) = X, ∀h ∈ H.

Proof. See proposition 2.1 of [1].

Theorem 4. If a be an invariant Riemannian metric on GH , and m be the

orthogonal complement of h in g, then there exists a bijection between the set

of all invariant Randers metrics on GH with the underlying Riemannian metric

a and the set

V1 = X ∈ m|Ad(h)X = X,< X,X >< 1, ∀h ∈ H

97

and the Randers metric F is of Berwald type if and only if (adX)m is skewadjoint and

a(X, [m,m]m) = 0

where(adX)m : m→ m, (adX)m(y) = [X, y]m

Proof. For the proof see [2]

Az an example let G be a Lie group, then G can be viewed as a reductive ho-mogeneous manifold with H = e and m = g, fix a left invariant Riemannianmetric a on G, then we have

V = X ∈ m|Ad(h)(X) = X, ∀h ∈ H

For any X ∈ V there corresponds a left-invariant Randers metric on G.we can also consider the right-invariant and even bi-invariant Randers metricon Lie groups.

3 Bi-invariant Randers metric on Lie group

some lie groups may posses a Randers metric which is invariant not only underleft translation but also under right translation.

Theorem 5. Let G be a Lie group with a bi-invariant Randers metric Fdefined by the Riemannian metric aijdx

i ⊗ dxj and the vector field X. Thenthe Randers metric F is of Berwald type.

Proof. let F (p, y) =√ap(y, y) + ap(X, y), now for s, t ∈ R

F 2(y + su+ tv) = a(y + su+ tv, y + su+ tv) + a2(X, y + su+ tv)

+ 2√a(y + su+ tv, y + su+ tv)a(X, y + su+ tv).

by definition

gy(u, v) =1

2

∂2

∂s∂t[F 2(y + su+ tv)]|s = t = 0

we have

gy(u, v) = a(u, v) + a(X,u)a(X, v) +a(u, v)a(X, y)√

a(y, y)

− a(v, y)a(u, y)a(X, y)

a(y, y)√a(y, y)

+a(X, v)a(u, y)

a(y, y)+a(X,u)a(v, y)√

a(y, y)(∗)

98

so for y, z ∈ g

gy(y, [y, z]) = a(y, [y, z]) + a(X, y)a(X, [y, z])

+a(y, [y, z])a(X, y)√

a(y, y)+ a([X, [y, z])

√a(y, y)

= a(y, [y, z])(1 +a(X, y)√a(y, y)

) + a(X, [y, z])(a(X, y) +√a(y, y)).

hence

gy(y, [y, z]) = a(y, [y, z])(F (y)√a(y, y)

) + a(X, [y, z])F (y). (∗∗)

since a is bi-invariant,a(y,[y,z]) = 0 and ad(x) is skew adjoint for every x ∈ g.since F is bi invariant gy(y, [y, z]) = 0 so from(*) we get a(X, [y, z]) = 0therefore, (G,F) is Berwald type.

References

[1] S. Deng And Z. Hou Invariant Randers metric on homogeneous Riemannian man-

ifols, 37(2004), J. Phys. A: Math Gen. 4353–4361.

[2] Dariush Latifi Bi-Invariant Randers metric on Lie group, Publ. Math. Debrecen

45–58 (2009).

[3] D. Bao, S. S. Chern and Z. Shen, An introduction to riemann-finsler geometry,

Springer-Verlag, New York 2000.

[4] D. Latifi Homogeneous geodesics in homogeneous Finsler space, J. Geom. Phys.

(2007), 1421-1433.

99



Generalized Vandermonde polynomial

Elhameh Rezaie1

The theory of symmetric polynomials is one of the most classical parts ofalgebra. A polynomial q(x1, · · · , xm) is said to be symmetric, if for any per-mutation σ ∈ Sm, we have

q(xσ(1), · · · , xσ(m)) = q(x1 · · · , xm).

a polynomial q(x1, · · · , xm) is said to be anti-symmetric, if for any permutationσ ∈ Sm, we have

q(xσ(1), · · · , xσ(m)) = ε(σ)q(x1 · · · , xm),

where ε(σ) is the sign of σ.

Definition. Let q ∈ Hd[x1, ..., xm]. Then we set

q∗ = T (G,χ)(q)

and we call it a symmetrized polynomial with respect to G and χ. Some times,we use the notation q∗(G,χ) instead of q. As a special case, if α ∈ Γ+

(m,d) we

denote the symmetrized monomial (Xα) by Xα,∗. Clearly

Hd(G,χ) = (Xα,∗ : α ∈ Γ+(m,d)),

where (set of vectors) denotes the subspace generated by a given set of vectors.

Definition. The Vandermonde series of G with respect to χ is defined recur-sively

G0 = G

Gr = < (ij) ∈ Gr − 1 : χ(ij) 6= (−1)r − 1 >

It is easy to see that the Vandermonde series of G with respect to χ is a sub-normal series. This series begins with G and ends with Gn = 1 and it mayhave some equal terms.

[email protected]

100

Definition. Let the vandermonde series of G with respect to χ be

1 = Gn Gn − 1 · · ·G1 G0 = G.

Then the relative Vandermonde polynomial of G with respect to χ is

V Gχ (x1, ..., xm) =

n−1∏

i−1(ij)∈Gr

(Xi −Xj).

Note that in the inner∏, we assume that i < j.

Theorem. Let the Vandermonde series of G with respect to χ be

1 = Gn Gn − 1 · · ·G1 G0 = G.

Suppose n ≥ 2. Then every element ofHd(G,χ) is a product of VGχ (x1, · · · , xm)

and a symmetric polynomial with respect to Gn − 1.

Example. we account the relative vandermnde polynomial for D6 with χ1, χ2

a =

(1 2 · · · m− 1 m2 3 · · · m 1

)m=3−−−→ a =

(1 2 32 3 1

)= (1, 2, 3)

b =

(1 2 · · · m− 1 m1 m · · · 3 2

)m=3−−−→ b =

(1 2 31 3 2

)= (2, 3)

G0 = D6 = 1, a, a2, b, ab, a2b1 = ξ , a = (1, 2, 3) , b = (2, 3)

a2 = (1, 3, 2) , ab = (1, 2) , a2b = (1, 3)

G1 = < (i, j) ∈ G0;χ1(i, j) 6= 1 >

G1 = < b ∈ G0 , ab ∈ G0 , a2b ∈ G0 ;χ1(b) = χ1(ab) = χ1(a

2b) = 1 >= 1

G1 = 1G0 = D6

V χ1

D6(x1, · · · , xm) =

n−1∏

r=1

∏

(i,j)∈Gr

(xi − xj) = 0

so the relative Vander mond polynomial is zero, now for grop D6, χ2

D6 = G0 = 1, a, a2, b, ab, a2b

101

−→ G1 =< ab, b, a2b ∈ G0;χ2(a) = χ2(ab) = χ2(a2b) 6= 1 >

−→ G2 =< ab, b, a2b ∈ G1;χ2(a) = χ2(ab) = χ2(a2b) = −1 >= 1

G2 = 1G1 = G0

V χ2

D6(x1, · · · , xm) =

n−1∏

r=1

∏

(i,j)∈G1

(xi − xj) = (x2 − x3)(x1 − x2)(x1 − x3)

= x2x21 − x22x1 + x1x

23 − x21x3 + x3x

22 − x23x2

Theorem. The relative Vandermonde polynomial is zero for dihederal groupD2n if n > 5.Proof. Note that permutation representation of G = Dn is obtained fromgeometric motions of the regular polygon with n edges. The geometric sym-metries of the regular polygon are rotations and mirror symmetries. Note thatno non-trivial rotation has fixed point so rotations are not transpose. D2n theother hand any mirror symmetry of n-gon changes at least 4 points if n > 5,so it is not a transpose. Therefore D2n does not contain any transpose and sothe vandermonde polynomial is zero.

References

[1] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Com-pany,(1974).

[2] H.F. da Cruz, J.A. Dias da Silva, Equality of immanantal decomposabletensors, Linear Algebra Appl. , 401 (2005) 29-46.

[3] H.F. da Cruz, J.A. Dias da Silva, Equality of immanantal decomposabletensors, II, Linear Algebra Appl. , 395 (2005) 95-119.

[4] J.A. Dias da Silva, Flags and equality of tensors, Linear Algebra Appl,232 (1996) 5575.

[5] J.A. Dias da Silva, Flags and equality of tensors, Linear Algebra Appl,342 (2002) 7991.

[6] J.A. Dias da Silva, Maria M. Torres,On the orthogonal dimension of or-bital sets, Linear Algebra Appl, 401 (2005) 77107.

102

[7] A. Fonseca,On the equality of families of decomposable symmetrized ten-sors, Linear Algebra Appl, 293 (1999) 114.

[8] I.G. Macdonald,Symmetric Functions and Orthogonal Polynomials,American Math, Soc., 1998.

[9] R. Merris,An identity for matrix functions, Pacific J. Math, 50 (1974)557562.

[10] R. Merris,Multilinear Algebra, Gordon and Breach Science Publisher,1997.

[11] M.R. Pournaki,On the orthogonal basis of the symmetry classes of ten-sors associated with certain characters,Linear Algebra Appl, 336 (2001)255260.

[12] B. Sagan,The Symmetric Group: Representation, Combinatorial Algo-rithms and Symmetric Functions, Wadsworth and Brook/Cole Math, Se-ries, 1991.

[13] M.A. Shahabi, K. Azizi, M.H. Jafari,On the orthogonal basis of symmetryclasses of tensors, J. Algebra , 237 (2001) 637646.

[14] . Shahryari,On the orthogonal bases of symmetry classes, J. Algebra 220(1999) 327332.

[15] M. Shahryari, M.A. Shahabi,On a permutation character of Sm, Linearand Multilinear Algebra, 44 (1998) 4552.

103

Second Workshop on Algebra and its Applications


A note on quasikernels of the irreducible charactersin finite p-groups

A. Saeidi1

Abstract

Let χ be an irreducible character of a finite group G. The set Z(χ) =x ∈ G : |χ(x)| = χ(1) is called the quasikernel of χ. It is easy to see thatZ(χ) is a normal subgroup of G, containing kerχ. Also, Z(χ)/ kerχ =Z(G/ kerχ) and the character χ is linear if and only if Z(χ) = G. In thistalk, we consider the set QKern(G) of all irreducible quasikernels of Gand obtain many results about this set. In particular, we show that if Gis a p-group, then G contains the extreme character degrees if and onlyif QKern(G) is a singletone.

1 Introduction

Throughout the paper, we only consider finite groups and complex characters.Irr(G) is the set of the irreducible characters of G. For χ ∈ Irr(G), set Z(χ) =x ∈ G : |χ(x)| = χ(1). Then Z(χ) is called the quasikernel of χ. It is easy tosee that Z(χ) is a normal subgroup of G, containing kerχ. Also Z(χ)/ kerχ =Z(G/ kerχ) [3, Lemma 2.27]. The character χ is linear if and only if Z(χ) = G.We call a normal subgroup N of G a nonlinear quasikernel if N = Z(χ) forsome χ ∈ Irr1(G). Nonlinear kernels are defined analogously. Denote byKern(G) and QKern(G), the set of the nonlinear kernels and quasikernels anon-abelian group G, respectively. We will frequently use the following termsand notations of [1]: the subgroups H and K of G are incident if either H ⊆ Kor K ⊂ H. A group G is called a J-group (respectively QJ-group) if each pairof distinct members of Kern(G) (respectively QKern(G)) are non-incident.Also G is a J0-group if |Kern(G)| = 1. We define a QJ0-group to be a groupwith |QKern(G)| = 1.

2 preliminaries

Lemma. Let G be a group. Then Z(G) =⋂

Z (χ) |χ ∈ Irr1(G).1Kharazmi University- [email protected]

104

Lemma. LetG be a group andNG. Assume that χ ∈ Irr(G) andN ≤ kerχ.If χ is the corresponding character of χ in G = G/N , then Z(χ) = Z(χ).

Lemma. [1, Lemma 29.5] G be a non-abelian nilpotent group. Then G is aJ-group if and only if it is a p-group with G′ ≤ Z(G) and expG′ = p.

Lemma. Let χ, ψ ∈ Irr(G). If kerχ ≤ kerψ, then Z(χ) ≤ Z(ψ).

3 Main Results

Proposition. Let G be a nilpotent QJ-group. Then G is a J-group.

Example. Let S be a non-cyclic simple group. Then G = S × Zp, where p isa prime number, is a QJ-group. However, it is routine to see that G is not aJ-group (we may directly use GAP [2] to verify this fact for S = A5 and p = 2,as the smallest examle of this type). This example shows that the previousproposition is not true at least for non-solvable groups.

Theorem. Let G be a p-group. Then G is a QJ0-group if and only if cd(G) =1, |G : Z(G)|1/2

.

Theorem. Let G be a p-group of class 2. Assume that G has some non-faithful nonlinear irreducible characters, all of which are of equal degree, sayd. Then G has no faithful irreducible characters. In particular, cd(G) = 1, d.

References

[1] Y. Berkovich and E. M. Zhmud´, Characters of Finite Groups, Part 2, Trans-lations of Mathematical Monographs, 181. AMS, Providence, RI, 1999.

[2] GAP groups, Algorithms, and Programming, Version 4.4.10, 2007.

[3] I.M. Isaacs, Character Theory of Finite Groups, Dover, New York, 1994.

105



Multiplication Ideals in Γ-rings

A. A. Estaji,1 A. Saghafi Khorasani∗2 and S. Baghdari3

Abstract

In this paper we shall develop a structure theory for multiplicativelyideal in rings. We introduce the notion multiplication ideal in Γ-rings.Let M be a commutative Γ-ring with unit and let N be faithful ideal ofΓ-ring M . We prove that N is a multiplication ideal if and only if1. For every non-empty collection of ideals Iλλ∈Λ ofM ,

⋂λ∈Λ(IλΓN) =

(⋂

λ∈Λ Iλ)ΓN and2. For every ideal K of M and ideal L of M such that K ⊂ LΓN thereexists an ideal J of M with J ⊂ L and K ⊆ JΓN .

Also, for ideal C in commutative Γ-ring with unit M , the prime rad-ical P (C) is the set of all x ∈ M which there exists n ∈ N such thatxγ0xγ0 · · · γ0x (n time) belongs to C. We show that for ideals A,Bin commutative Γ-ring with unit M and faithful multiplication idealW ⊆M , AΓW ⊆ BΓW if and only if either A ⊆ B orW = ([B : A])ΓW .SΓ is introduced and we prove that for ideals A,B in commutative Γ-ringwith unit M and W ∈ SΓ, A = B if and only if AΓW = BΓW .

References

[1] W. E. Barnes, On the Γ-rings of Nobusawa, Pacific J. Math. 18 (1966), 411-422.

[2] S. Kyuno, On the radicals of Γ-rings, Osaka J. Math. 12 (1975) 639-645.

[3] S. Kyuno, On prime gamma rings , Pacific J. Math. 75 No. 1 (1978) 185-190.

[4] G. H. Low and P. F. Smith, Multification Modules and ideals, Comm. Algebra

(1990) 4353 - 4375.

[5] N. Nobusawa, On a generalization of the ring theory, Osaka J. Math. 1 (1964),

81-89.

1Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,Iran, aa [email protected]

2Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,Iran, [email protected]

3Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,Iran, [email protected]

106



2-absorbing Submodules of Multiplication Modules

F. Soheilnia1

Abstract

In this talk, all rings are commutative with nonzero identity and allmodules are considered to be unitary. Prime submodules have an im-portant role in the module theory over commutative rings. We recallthat a proper submodule N of R-module M is called a prime submoduleif whenever a ∈ R and m ∈ M with am ∈ N , then either m ∈ N ora ∈ (N :R M).

Badawi in [1] generalized and studied the concept of prime ideals ina different way. He defined a nonzero proper ideal I of R to be a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then eitherab ∈ I or ac ∈ I or bc ∈ I. This concept has a generalization calledweakly 2-absorbing ideals, which has been studied in [2].

This talk is based on the papers [3] and [4]. In this note, we gener-alize the two concepts 2-absorbing ideals and weakly 2-absorbing idealsto submodules of a module over a commutative ring as generalizations ofprime submodules. We define a proper submodule N to be a 2-absorbing(resp,. weakly 2-absorbing) submodule of M if whenever a, b ∈ R andm ∈M with abm ∈ N (resp,. 0 6= abm ∈ N), then either ab ∈ (N :R M)or am ∈ N or bm ∈ N . We give some basic properties of these classes ofsubmodules in multiplication modules.

References

[1] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc,

75 (2007), 417-429.

[2] A. Badawi and A. Yousefian Darani, On weakly 2-absorbing ideals of commutative

rings, to appear.

[3] A. Yousefian Darani and F. Soheilnia, 2-absorbing and weakly 2-absorbing submod-

ules, Thai J. Math, 9 (3) (2011), 577-584.

[4] A. Yousefian Darani and F. Soheilnia, On n-absorbing submosules, to appear.

1Department of Mathematics and Applications, University of Mohaghegh Ardabili,[email protected]

107



Ideals of Novikov Algebras on Lie Algebras

H. Tayer1

Abstract

We study ideals of Novikov algebras and Novikov structures on finit-dimensional Lie algebras. We show the example of Lie algebra whichadmit a Novikov structure, present some structure theory concerningideals in Novikov algebras.

References

[1] D. Burde, K. Dekimpe and K. Vercammen, Novikov algebras and Novikov struc-

tures on Lie algebras, J. Linear Algebra and its Applications, 429(2008), 31-41.

[2] E. I. Zelmanov, On a class of local translation invariant Lie algebras, Sov. Math.

Dokl. 35(1987), 216-218.

[3] X. Xu, Classifications of simple Novikov algebras and their irreducible modules of

characteristic 0, J. Algebra, 246(2001), 673-707.

1Shahed University, [email protected]

108



Leibniz Algebras Admitting non–degeneratedDerivations

C. Zargeh1

Abstract

The main aim of this paper is to introduce notion of Leibniz algebrasand reprove the main theorem in which any complex Leibniz algebraadmitting a non–degenerated derivation is nilpotent. Also we give moredetails related to proof of main theorems.

1 Introduction

Leibniz algebras were first Introduced by Loday in [5, 6] as a non–antisymmetricversion of Lie algebras. Many results of Lie algebras were also established inLeibniz algebras. For instance, the classical results on Cartan Subalgebras,regular elements and others from the theory of Leibniz algebras. [1]In 1955, Jacobson [3] proved that every Lie algebra over a field of characteris-tic zero admitting a non singular derivation is nilpotent. The same work hasbeen done by Bajo [2] for non–singular prederivations and several example ofnilpotent Lie algebras, whose pre–derivations are nilpotent has been provided.In this paper we review a well–known theorem of Jacobson [3] states thatany Lie algebra admitting a non–singular derivation must be a nilpotent Liealgebra, and then prove the analogue result for Leibniz algebras.

2 Preliminaries

We present some known notions and results related to Leibniz algebras thatwe use further in this paper.

Definition 1.([5]) An algebra L over a field F is called Leibniz algebra if forany x, y, z ∈ L the Leibniz identity

[[x, y], z] = [[x, z], y] + [x, [y, z]]

1Sama Technical and Vocational Training College, Islamic Azad University, Mahabadbranch, Mahabad, Iran, [email protected]

109

is satisfied, where [−,−] is the multiplication in L. In other words, the rightmultiplication operator [−, z] by any element z is a derivation.Any Lie algebra is a Leibniz algebra, and conversely any Leibniz algebra L isa Lie algebra if [x, x] = 0 for all x ∈ L. More ever, if Lann = ideal〈[x, x]|x ∈ L〉then factor algebra LLann is a Lie algebra.For a Leibniz algebra L consider the following derived and lower central series:

(i) L(1) = L L(n+1) =[L(n), L(n)

]n > 1

(ii) L1 = L L(n+1) = [Ln, L] n > 1

Definition 2.([5]) An algebra L is called solvable (nilpotent) if there exists ∈ N(k ∈ N) such that L(s) = 0(Lk = 0, respectively)

Theorem 3. Let A be a linear transformation of vector space V . Then Vdecomposes into the direct sum of characteristic subspaces V = Vλ1 ⊕· · ·⊕Vλk

with respect toA, where Vλi=

x ∈ V |(A− λiI)

k(x) = 0 for some k ∈ N

and λi 1 ≤ i ≤ k, are eigenvalues of A.Now we define derivation of Leibniz algebras.

Definition 4. A linear operator d : L −→ L is called a derivation of L if

d([x, y]) = [d(x), y] + [x, d(y)] for any x, y ∈ L.

If x be an arbitrary element of L, we consider the right multiplication operatorRx : L −→ L defined by Rx(z) = [z, x]. Right multiplication operators arederivations of the algebra L. The set R(L) = Rx|x ∈ L is a Lie algebra withrespect to the commutator and the following identity holds:

RxRy −RyRx = R[y,x]

Definition 5.([3]) A subset S of an associative algebra A over a field F iscalled weakly closed if for every pair (a, b) ∈ S × S an element γ(a, b) ∈ F isdefined such that ab+ γ(a, b)ba ∈ S.

Theorem 6. [Engel’s theorem] A Leibniz algebra L is nilpotent if and onlyRx is nilpotent for any X ∈ L.

3 Main Theorems

In this section we reprove known results of Lie algebras for Leibniz algebrasand give more details.

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Lemma 7. Let L be a finite dimensional Leibniz algebra with a derivation ddefined on it and L = Lρ1⊕· · ·⊕Lρs be a decomposition of L into characteristicspaces with respect to d. Then for any α, β ∈ spec(d) we have:

[Lα, Lβ ] ⊆Lα+β , if α+ β is an eigenvalue of d0, if α+ β isnot an eigenvalue of d

We have the similar result for automorphism of Leibniz algebras where:

[Lα, Lβ ] ⊆Lαβ , if αβ is an eigenvalue of A0, if α+ β isnot an eigenvalue of A,

(A is an automorphism).

Theorem 8.([4]) Let L be a finite dimensional complex Leibniz algebra whichadmits a non–degenerated derivation. Then L is a nilpotent algebra.

Proof. Let d be a non–singular derivation of a Leibniz algebra L. Due to d wedecompose L into characteristic spaces Lρ1 , · · · , Lρn such that

L = Lρ1 ⊕ · · · ⊕ Lρn

If α, β ∈ spec(d), then by 3 we have:[· · · [Lα, Lβ ], Lβ , · · · , Lβ︸︷︷︸

]⊆ Lα+kβ

for sufficiently large k ∈ N, α+kβ /∈ spec(d), so according to the lower centralseries definition we have:

[· · · [Lα, Lβ ], Lβ , · · · , Lβ ] = 0

So any right multiplication operatorRx (x ∈ Lβ) is nilpotent. Since α, β weretaken arbitrarily, it follows that every operator from Uk

i=1R(Lρi) is nilpotent.On the other hand we know that

RxRy −RyRx = R[y,x]

therefore if we consider γ = −1, then we can say that Uki=1R(Lρi) is weakly

closed set of R(L). Thus R(L) as an enveloping of R(Lρi) is nilpotent. Finallyby Engel’s Theorem we obtain the result, i. e, L is nilpotent.

Theorem 9.([4]) If L be a complex Leibniz algebra admitting an automor-phism of prime order with no fixed–points, then L is nilpotent.

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Proof. Let A be an automorphism of Leibniz algebra L. Since A has no fixed–points then 1 is not an eigenvalue of A.Let L = Lρ1 ⊕ · · · ⊕ Lρk a decomposition of L into characteristic spaces withrespect to A. Due to the fact that A is an automorphism of prime orderwe know that the spectrum of A include primiting p–th roots of the unity.Therefore, for any α, β ∈ space(A) there exist k ∈ N such that αβ = 1 /∈space(A). Hence by similar Lemma to lemma 3 we obtain

[· · · [Lα, Lβ ], Lβ ] , · · · , Lβ ] ⊆ Lαβk = 0

Thus, for x ∈ Lβ any Rx is nilpotent, and similarly as in the previous proofof theorem 3, L is nilpotent.

References

[1] S. A. Albeverio, S. A. Ayupov, B. A. Omirov, Cartan Sub algebras, weight spaces,

and criterion of soluability of finite dimensional Leibniz algebras, Rev. Mat. Complut.

19(1)(2006), 183-195.

[2] I. Bajo, Lie algebras admitting non-singular prederivations, Indag. Mathem, 8(4),

(1997), 433-437.

[3] N. Jacobson, A note on automorphisms and derivations of Lie algebras , Proc.

Amer. Math. Soc. 6(1955), 281-283.

[4] M. Larda, I. M. Rikhsiboev, R. M. Turdibave, Automorphisms and derivations of

Leibniz algebras, arXIv: 1103. 4721v1, (2011), 12 p.

[5] J. -L. Loday, Cyclic homology, Grundl. Math. Wiss. Bd. 301, Springer-Verlag,

Berlin, (1992).

[6] J. -L. Loday, Une version non commutative des algebras de Lie: les algebres de

Leibniz, Enseign. Math. (2)39(1993), 269-293.

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Total Coloring of Powers and Fractional Powers of aGraph

M. S. Najafian1 and M. Zavieh∗2

Abstract

The coloring problem in graph theory is the result of an optimizationproblem. Vizing (1964) and Behzad (1965) independently conjecturedthat the total chromatic number of a simple graph G (χT (G) or χ

′′(G))never exceeds ∆(G) + 2. Beside, it is clear that the minimum number ofcolors needed is ∆(G) + 1, since the vertex having maximum degree andall its incident edges must receive distinct colors. If the total coloringconjecture is true, then the total chromatic number of a graph is alwayseither ∆(G) + 1, or ∆(G) + 2.

Graph fractional powers have been introduced in [1] and a relation hasbeen stated between graph fractional powers and total graph as follows:If G is a simple graph then χT (G) = χ(G

22 ) [1]. In this paper, we present

a proof for above problem using graph isomorphism. Finally, we referto some theorems and one conjecture about total coloring of powers ofcycles which have been considered in [2].

References

[1] M. N. Iradmusa, On colorings of graph fractional powers, Discrete Mathematics,

310(2010), 1551-1556.

[2] C. N. Campos and C. P. de Mello, A result on the total colouring of powers of

cycles, Discrete Mathematics, 18(2004), 47-52.

1Department of Mathematical Sciences, Zanjan University, Zanjan, Iran, [email protected]

2Department of Mathematical Sciences, Zanjan University, Zanjan, Iran,mehdi [email protected]

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List of Participants

Yousef Abbaspour, University of Mohaghegh ArdabiliJavad Abdali, Islamic Azad University, Khoy BranchReza Abdolmaleki, Shahed UniversityHossein Abdolzadeh, University of Mohaghegh ArdabiliSeyyed Mohammad Ajdani, Islamic Az. Univ., Science and Research BranchLeyla Alipour, Urmia UniversityMohammad Hassan Anjom Shoa, University of BirjandJafar Azami, University of Mohaghegh ArdabiliAyman Badawi, American University of Sharjah, United Arab EmiratesAnahita Bandebahman, University of IsfahanAbdollah Borhanifar, University of Mohaghegh ArdabiliSoheila Ebrahimian, University of Mohaghegh ArdabiliMajid Eghbali, Islamic Azad University, Soofian BranchAmir Ehsani, Islamic Azad University, Mahshahr BranchFarkhondeh Farzalipour, Payame Noor UniversityFatemeh Farzalipour, University of Mohaghegh ArdabiliSamira Fathiye, University of Mohaghegh ArdabiliGholamhossein Fathtabae, University of KashanMasoud Ganji, University of Mohaghegh ArdabiliJuergen Garloff, University of Konstanz, GermanyGhader Ghasemi, University of Mohaghegh ArdabiliPeyman Ghiasvand, Payame Noor UniversityMahboobeh Ghorbanalizadeh, University of Mohaghegh ArdabiliHanieh Golmakani, Islamic Azad University, Mashhad BranchHassan Haghighi, K. N. Toosi University of TechnologyKazem Haghnejad Azar, University of Mohaghegh ArdabiliHassan Hosseinzadeh, Islamic Azad University, Science and Research BranchAdel Kazemi-Piledaraq, University of Mohaghegh ArdabiliAhmad Khojali, University of Mohaghegh ArdabiliHassan Khodaeimehr, Amirkabir University of TechnologyJanos Kurdics, Nyıregyhaza, HungaryShokufeh Lotfi, University of ZanjanNasrin Malekzadeh, University of Mohaghegh ArdabiliSadegh Mohammadikhah, Payame Noor UniversityElahe Mohammadzadeh, Payame Noor UniversityFahime Mohammadzadeh, Payame Noor UniversityMohammad Zarebnia, University of Mohaghegh ArdabiliNazila Moosavi, University of Mohaghegh Ardabili

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Moharram Nejad Iradmusa, Sharif University of TechnologyJavad Noori, Islamic Azad University, Kaleibar BranchBehnaz Pahlavsay, University of Mohaghegh ArdabiliSafar Parsi, University of Mohaghegh ArdabiliShirooye Payrovi, Imam Khomeini International UniversityNaser Pourreza, Islamic Azad University, Tehran BranchNahid Rafi’ei, University of Mohaghegh ArdabiliRahim Rahmatiasghar, University of TehranMajid Rahrozargar, Kharazmi UniversityZari Ranjbar, University of Mohaghegh ArdabiliJabraeil Razmjou, University of Mohaghegh ArdabiliElhameh Rezaei, Islamic Azad University, Kaleibar BranchNavid Rezaei Melal, University of ZanjanAmin Saeidi, Kharazmi UniversityAli Saghafi Khorasani, Hakim Sabzevari UniversityShahram Motmaen, University of Mohaghegh ArdabiliFatemeh Soheilnia, University of Mohaghegh ArdabiliVali Soleymani, University of Mohaghegh ArdabiliAli Soleymanjahan, University of KordestanParisa Solhi, University of Mohaghegh ArdabiliGolsa Tarverdizadeh, University of Mohaghegh ArdabiliHanie Tayer, Shahed UniversitySiamak Yassemi, University of TehranAhmad Yousefian Darani, University of Mohaghegh ArdabiliChia Zargeh, Islamic Azad University, SamaMehdi Zavieh, University of Zanjan

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Second Seminar on Algebra and its Applications · Second Seminar on Algebra and its Applications...

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