Andreas Bernhard Zeidler

Abstract Algebra

Rings, Modules, Polynomials, Ring Extensions,

Categorial and Commutative Algebra

December 23, 2019 (875 pages)

考え

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This book is dedicated to the entire mathematical society: to allthose who contribute to mathematics and keep it alive by teaching it.

Copyright (C) December 23, 2019

by Andreas Bernhard Zeidler

This material may be referenced freely, as long as the original author(s) anda reference to the site of this document is clearly stated. It may be dis-tributed only subject to the terms and conditions set forth in the Open Pub-lication License, v1.0 or later, given in section 0.2 here (the latest version ispresently available at http://www.opencontent.org/openpub). Distributionof the work or derivative of the work for commercial purposes is prohibited,unless prior permission is obtained from the copyright holder.

Contents

0 Prelude 5

0.1 About this Book . . . . . . . . . . . . . . . . . . . . . . . . . 5

0.2 Open Publication License . . . . . . . . . . . . . . . . . . . . 9

0.3 Notation and Symbols . . . . . . . . . . . . . . . . . . . . . . 10

0.4 A Primer on Sets . . . . . . . . . . . . . . . . . . . . . . . . . 16

0.5 A Primer on Classes . . . . . . . . . . . . . . . . . . . . . . . 29

1 Groups and Rings 34

1.1 Defining Groups . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.3 Defining Rings . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.5 First Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 62

1.6 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

1.7 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 81

1.8 Ordered Rings . . . . . . . . . . . . . . . . . . . . . . . . . . 89

1.9 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2 Commutative Rings 100

2.1 Maximal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 100

2.2 Prime Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

2.3 Radical Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.4 Noetherian Rings . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.5 Unique Factorisation Domains . . . . . . . . . . . . . . . . . . 123

2.6 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . 135

2.7 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . 142

2.8 Lasker-Noether Decomposition . . . . . . . . . . . . . . . . . 148

2.9 Finite Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

2.10 Localization of Rings . . . . . . . . . . . . . . . . . . . . . . . 159

2.11 Local Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

2.12 Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . . 177

3 Modules 183

3.1 Defining Modules . . . . . . . . . . . . . . . . . . . . . . . . . 183

3.2 First Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 196

3.3 Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

3.4 Ideal-induced Modules . . . . . . . . . . . . . . . . . . . . . . 210

3.5 Block Decompositions . . . . . . . . . . . . . . . . . . . . . . 214

3.6 Dependence Relations . . . . . . . . . . . . . . . . . . . . . . 220

3.7 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . 224

3.8 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 230

3.9 Isomorphism Theorems . . . . . . . . . . . . . . . . . . . . . 237

3.10 Rank of Modules . . . . . . . . . . . . . . . . . . . . . . . . . 245

3.11 Noetherian Modules . . . . . . . . . . . . . . . . . . . . . . . 252

3.12 Localization of Modules . . . . . . . . . . . . . . . . . . . . . 262

3.13 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . 271

4 Linear Algebra 289

4.1 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 289

4.2 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . 299

4.3 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . 305

4.4 Matrix Representations . . . . . . . . . . . . . . . . . . . . . 316

4.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

4.6 Rank of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 339

4.7 Duality Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 350

4.8 Vector Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 368

4.9 Linear Optimization . . . . . . . . . . . . . . . . . . . . . . . 381

5 Structure Theorems 382

5.1 Primary Decomposition . . . . . . . . . . . . . . . . . . . . . 382

5.2 Invariant Factors . . . . . . . . . . . . . . . . . . . . . . . . . 392

5.3 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 406

5.4 Jordan Normal Form . . . . . . . . . . . . . . . . . . . . . . . 416

5.5 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . 416

6 Functional Analysis 417

6.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

6.2 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 418

6.3 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 418

6.4 Operatoralgebras . . . . . . . . . . . . . . . . . . . . . . . . . 418

7 Polynomial Rings 419

7.1 Monomial Orders . . . . . . . . . . . . . . . . . . . . . . . . . 419

7.2 Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 425

7.3 Defining Polynomials . . . . . . . . . . . . . . . . . . . . . . . 431

7.4 The Standard Cases . . . . . . . . . . . . . . . . . . . . . . . 437

7.5 Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 445

7.6 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

8 Polynomials in One Variable 446

8.1 Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . 446

8.2 Irreducibility Tests . . . . . . . . . . . . . . . . . . . . . . . . 446

8.3 The Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . 446

8.4 Polynomials of Low Degree . . . . . . . . . . . . . . . . . . . 446

9 Ring Extensions 447

9.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

9.2 Integral Extensions . . . . . . . . . . . . . . . . . . . . . . . . 449

9.3 Noetherian Normalisation . . . . . . . . . . . . . . . . . . . . 449

9.4 Krull Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 449

9.5 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

10 Group Theory 45110.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 45110.2 Group Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 45110.3 Representation Theory . . . . . . . . . . . . . . . . . . . . . . 456

11 Homological Algebra 45711.1 Diagram Chasing . . . . . . . . . . . . . . . . . . . . . . . . . 45811.2 Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . 45811.3 Flat Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 45811.4 Homology Modules . . . . . . . . . . . . . . . . . . . . . . . . 45811.5 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

12 Categorial Algebra 45912.1 Categories and Functors . . . . . . . . . . . . . . . . . . . . . 45912.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45912.3 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46012.4 Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . 460

13 Graded Rings 46113.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

14 Valuations 462

15 Symbols 463

16 List of Proofs 465

17 Proofs - Fundamentals 466

18 Proofs - Rings and Modules 503

19 Proofs - Commutative Algebra 741

Chapter 0

Prelude

0.1 About this Book

The Aim of this BookMathematics knows two directions - analysis and algebra - and any math-ematical discipline can be weighted how analytical resp. algebraical it is.Analysis is characterized by having a notion of convergence that allows toapproximate solutions (and reach them in the limit). Algebra is character-ized by having no convergence and hence allowing finite computations only.This book now is meant to be a thorough introduction into algebra.

Likewise every textbook on mathematics is drawn between two pairs ofextremes: (easy understandability versus great generality) and (complete-ness versus having a clear line of thought). Among these contrary poles weusually chose (generality over understandability) and (completeness over aclear red line). Nevertheless we try to reach understandability by being veryprecise and accurate and including many remarks and examples.

At last some personal philosophy: a perfect proof is like a perfect gem -unbreakably hard, spotlessly clear, flawlessly cut and beautifully displayed.In this book we are trying to collect such gemstones. And we are proud toclaim, that we are honest about where a proof is due and present completeproofs of almost every claim contained herein (which makes this textbookvery different from most others).

This Book is Written formany different kinds of mathematicians: primarily is meant to be a source ofreference for intermediate to advanced students, who have already had a firstcontact with algebra and now closely examine some topic for their seminars,lectures or own thesis. But because of its great generality and completenessit is also suited as an encyclopedia for professors who prepare their lecturesand researchers who need to estimate how far a certain method carries.Frankly this book is not perfectly suited to be a monograph for novices tomathematics. So if you are one we think you can greatly profit from thisbook, but you will probably have to consult additional monographs (at amore introductory level) to help you understand this text.

5

PrerequisitesWe take for granted, that the reader is familiar with the basic notions ofnaive logic (statements, implication, proof by contradiction, usage of quanti-fiers, . . . ) and naive set theory (Cantor’s notion of a set, functions, partiallyordered sets, equivalence relations, Zorn’s Lemma, . . . ). We will present ashort introduction to classes and the NBG axioms when it comes to cate-gory theory. Further we require some basic knowledge of integers (includingproofs by induction) and how to express them in decimal numbers. We willsometimes use the field of real numbers, as they are most probably well-known to the reader, but they are not required to understand this text.Aside from these prerequisites we will start from scratch.

Topics CoveredWe start by introducing groups and rings, immediately specializing on rings.Of general ring theory we will introduce the basic notions only, e.g. theisomorphism theorems. Then we will turn our attention to commutativerings, which will be the first major topic of this book: we closely studymaximal ideals, prime ideals, intersections of such (radical ideals) and therelations to localization. Further we will study rings with chain conditions(noetherian and artinian rings) including the Lasker-Noether theorem. Thiswill lead to standard topics like the fundamental theorem of arithmetic. Andwe conclude commutative ring theory by studying discrete valuation rings,Dedekind domains and Krull rings.

Then we will turn our attention to modules, including rank, dimensionand length. We will see that modules are a natural and powerful general-ization of ideals and large parts of ring theory generalizes to this setting,e.g. localization and primary decomposition. Module theory naturally leadsto linear algebra, i.e. the theory of matrix representations of a homomorph-ism of modules. Applying the structure theorems of modules (the theoremof Prüfer to be precise) we will treat canonical form theory (e.g. Jordannormal form).

Next we will study polynomials from top down: that is we introducegeneral polynomial rings (also known as group algebras) and graded alge-bras. Only then we will regard the more classical problems of polynomialsin one variable and their solvability. Finally we will regard polynomials inseveral variables again. Using Gröbner bases it is possible to solve abstractalgebraic questions by purely computational means.

Then we will return to group theory: most textbooks begin with thistopic, but we chose not to. Even though group theory seems to be elementaryand fundamental this is not quite true. In fact it heavily relies on argumentslike divisibility and prime decomposition in the integers, topics that arenative to ring theory. And commutative groups are best treated from thepoint of view of module theory. Never the less you might as well skip theprevious sections and start with group theory right away. We will presentthe standard topics: the isomorphism theorems, group actions (includingthe formula of Burnside), the theorems of Sylow and lastly the p-q-theorem.However we are aiming directly for the representation theory of finite groups.

The first part is concluded by presenting a thorough introduction to whatis called multi-linear algebra. We will study dual pairings, tensor productsof modules (and algebras) over a commutative base ring, derivations andthe module of differentials.

6 0 Prelude

Thus we have gathered a whole bunch of separate theories - and it is time fora second structurisation (the first structurisation being algebra itself). Wewill introduce the notions of categories, functors, equivalence of categories,(co-)products and so on. Categories are merely a manner of speaking -nothing that can be done with category theory could not have been achievedwithout. Yet the language of categories presents a unifying concept for allthe different branches of mathematics, extending far beyond algebra. So wewill first recollect which part of the theory we have established is what inthe categorical setting. The categorial language is the right setting to treatchain complexes, especially exact sequences of modules. Finally we willpresent the basics of abelian categories as a unifying concept of all thoseseparate theories.

We will then aim for some more specialized topics: At first we will studyring extensions and the dimension theory of commutative rings. A specialcase are field extensions including the beautiful topic of Galois theory. After-wards we turn our attention to filtrations, completions, zeta-functions andthe Hilbert-Samuel polynomial. Finally we will venture deeper into numbertheory: studying the theory of valuations up to the theorem of Riemann-Roche for number fields.

Topics not CoveredThere are many instances where, dropping a finiteness condition, one has tointroduce some topology in order to pursue the theory further. Examplesare: linear algebra on infinite dimensional vector-spaces, representation the-ory of infinite groups and Galois theory of infinite field extensions. Anothernatural extension would be to introduce the Zariski topology on the spec-trum of a ring, which would lead to the theory of schemes directly. Yet thescope of this text is purely algebraic and hence we will usually stop at thepoint where topology sets in (but give hints for further readings).

The Two PartsMathematics has a peculiarity to it: there are problems (and answers) thatare easy to understand but hard to prove. The most famous example isFermat’s Last Theorem - the statement (for any n ≥ 3 there are no non-trivial integers (a, b, c) ∈ Z3 that satisfy the equation an + bn = cn) canbe understood by anyone. Yet the proof is extremely hard to provide. Ofcourse this theorem has no value of its own (it is the proof that containsdeep insights into the structure of mathematics), but this is no general rule.E.g. the theorem of Wedderburn (every finite skew-field is a field) is easy anduseful, but its proof will be performed using a beautiful trick-computation(requiring the more advanced method of cyclotomic polynomials).

Thus we have chosen an unusual approach: we have separated the truth(i.e. definitions, examples and theorems) from their proofs. This enables usto present the truth in a stringent way, that allows the reader to get a feelfor the mathematical objects displayed. Most of the proofs could have beengiven right away, but in several cases the proof of a statement can only bedone after we have developed the theory further. Thus the sequel of theo-rems may (and will) be different from the order in which they are proved.Hence the two parts.

0.1 About this Book 7

Our Best AdviceIt is a well-known fact, that some proofs are just computational and onlycontain little (or even no) insight into the structure of mathematics. Othersare brilliant, outstanding insights that are of no lesser importance than thetheorem itself. Thus we have already included remarks of how the proofworks in the first part of this book. And our best advice is to read a sectionentirely to get a feel for the objects involved - only then have a look at theproofs that have been recommended. Ignore the other proofs, unless youhave to know about them, for some reason. At several occasions this textcontains the symbols (♦) and (�) . These are meant to guide the reader inthe following ways:

(♦) As we have assorted the topics covered thematically (paying little at-tention to the sequel of proofs) it might happen that a certain exampleor theorem is far beyond the scope of the theory presented so far. Inthis case the reader is asked to read over it lightly (or even skip itentirely) and return to it later (after he has gained some more experi-ence).

(�) On some very rare occasions we will append a theorem without givinga proof (if the proof is beyond the scope of this text). Such an instancewill be marked by the black box symbol. In this case we will alwaysgive a complete reference of the most readable proof the author isaware of. And this symbol will be hereditary, that is once we usea theorem that has not been proved any other proof relying on theunproved statement will also be branded by the black box symbol.

8 0 Prelude

0.2 Open Publication License

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0.2 Open Publication License 9

0.3 Notation and Symbols

ConventionsWe now wish to include a set of the frequently used symbols, conventionsand notations. In particular we clarify the several domains of numbers.

• First of all we employ the nice convention (introduced by Halmos) towrite iff as an abbreviation for if and only if.

• We denote the set of natural numbers - i.e. the positive integers in-cluding zero - by N := { 0, 1, 2, 3, . . . }. Further for any two integersa, b ∈ Z we denote the interval of integer numbers ranging from a tob by a . . . b := { k ∈ Z | a ≤ k ≤ b }.

• We will denote the set of integers by Z = N∪ (−N), and the rationalsby Q = { a/b | a, b ∈ Z, b 6= 0 }. Whereas Z will be taken for granted,Q will be introduced as the quotient field of Z.

• The reals will be denoted by R and we will present an example of howthey can be defined (without proving their properties however). Thecomplex numbers will be denoted by C = { a+ ib | a, b ∈ R } and wewill present several ways of constructing them.

• (♦) We will sometimes use the Kronecker-Symbol δ(a, b) (in the liter-ature this is also denoted by δa,b), which is defined to be

δ(a, b) = δa,b :=

{1 if a = b0 if a 6= b

In most cases a and b ∈ Z will be integers and 0, 1 ∈ Z will be integers,too. But in general we are given some ring (R,+, ·) and a, b ∈ R. Thenthe elements 0 and 1 ∈ R on the right hand side are taken to be thezero-element 0 and unit-element 1 of R again.

• We will write A ⊆ X to indicate that A is a subset of X and A ⊂ Xwill denote strict inclusion (i.e. A ⊆ X and there is some x ∈ X withx 6∈ A). For any set X we denote its power set (i.e. the set of allits subsets) by P(X) := {A | A ⊆ X }. And for a subset A ⊆ X wedenote the complement of A in X by CA := X \A.

• Listing several elements x1, . . . , xn ∈ X of some set X, we do not re-quire these xi to be pairwise distinct (e.g. x1 = x2 might well happen).Yet if we only give explicit names xi to the elements of some previ-ously given subset A = {x1, . . . , xn } ⊆ X we already consider the xito be pairwise distinct (that is xi = xj implies i = j). Note that if thexi (not the set {x1, . . . , xn }) have been given, then {x1, . . . , xn } mayhence contain fewer than n elements!

• Given an arbitrary set of sets A one defines the grand union⋃A and

the grand intersection⋂A to be the set consisting of all elements a

that are contained in one (resp. all) of the sets A ∈ A, formally⋃A := { a | ∃A ∈ A : a ∈ A }⋂A := { a | ∀A ∈ A : a ∈ A }

10 0 Prelude

Note that⋂A only is a well-defined set, if A 6= ∅ is non-empty. A well-

known special case of this is the following: consider any two sets A andB and let A := {A,B }. Then A ∪ B =

⋃A and A ∩ B =

⋂A. This

notion is just a generalization of the ordinary union and intersectionto arbitrary collections A of sets.

• If X and Y are any sets then we will denote the set of all functionsfrom X to Y by F(X,Y ) = Y X = { f | f : X → Y }. And for anysuch function f : X → Y : x 7→ f(x) we will denote its graph (notethat from the set-theoretical point of view f is its graph) by

Γ(f) := { (x, f(x)) | x ∈ X } ⊆ X × Y

• Once we have defined functions, it is easy to define arbitrary cartesianproducts. That is let I 6= ∅ be any non-empty set and for any i ∈ Ilet Xi be another set. Let us denote the union of all the Xi by X

X :=⋃i∈I

Xi = {x | ∃ i ∈ I : x ∈ Xi }

Then the cartesian product of the Xi consists of all the functionsx : I → X such that for any i ∈ I we have xi := x(i) ∈ Xi. Note thatthereby it is customary to write (xi) in place of x. Formally∏

i∈IXi := {x : I → X : i 7→ xi | ∀ i ∈ I : xi ∈ Xi }

• If I 6= ∅ is any index set and Ai ⊆ X is a non-empty Ai 6= ∅ subsetof X (where i ∈ I) then the axiom of choice states that there isa function a : I → X such that for any i ∈ I we get a(i) ∈ Ai. Inother words the product of non-empty sets is non-empty again. Itis a remarkable fact that this seemingly trivial property has far-flungconsequences, the lemma of Zorn being the most remarkable one.∏

i∈IXi 6= ∅ ⇐⇒ ∀ i ∈ I : Xi 6= ∅

• Let X 6= ∅ be a non-empty set, then a subset of the form R ⊆ X ×Xsaid to be a relation on X. And in this case it is customary to writexRy instead of (x, y) ∈ R. This notation will be used primarily forpartial orders and equivalence relations (see below).

• Consider any non-empty set X 6= ∅ again. Then a relation ∼ on X issaid to be an equivalence relation on X, iff it is reflexive, symmetricand transitive. Formally that is for any x, y and z ∈ X we get

x = y =⇒ x ∼ yx ∼ y =⇒ y ∼ x

x ∼ y, y ∼ z =⇒ x ∼ z

0.3 Notation and Symbols 11

And in this case we define the equivalence class [x] of x to be the setof all y ∈ X being equivalent to x, that is [x] := { y ∈ X | x ∼ y }. Andthe set of all equivalence classes is denoted by X/∼ := { [x] | x ∈ X }.Example: if f : X → Y is any function then we obtain an equivalencerelation ∼ on X by letting x ∼ y :⇐⇒ f(x) = f(y). Then theequivalence class of x ∈ X is just the fiber [x] = f−1(f(x)).

• Consider a non-empty set X 6= ∅ once more. Then a family of subsetsP ⊆ P(X) is said to be a partition of X, iff for any P , Q ∈ P weobtain the statements

X =⋃P

P 6= ∅P 6= Q =⇒ P ∩Q = ∅

Example: if ∼ is an equivalence relation on X, then X/∼ is a partitionofX. Conversely if P is a partition ofX, then we obtain an equivalencerelation ∼ on X by letting x ∼ y :⇐⇒ ∃P ∈ P : x ∈ P and y ∈ P .Hence there is a one-to-one correspondence between the equivalencerelations on X and the partitions of X given by ∼ 7→ X/∼.

• Consider any non-empty set I 6= ∅, then a relation ≤ on I is said to bea partial order on I, iff it is reflexive, transitive and anti-symmetric.Formally that is for any i, j and k ∈ I we get

i = j =⇒ i ≤ ji ≤ j, j ≤ k =⇒ i ≤ ki ≤ j, j ≤ i =⇒ i = j

And ≤ is said to be a total or linear order iff for any i, j ∈ I wealso get i ≤ j or j ≤ i (that is any two elements contained in I canbe compared). Example: for any set X the inclusion relation ⊆ is apartial (but not linear) order on P(X). If now ≤ is a linear order onI, then we define the minimum and maximum of i, j ∈ I to be

i ∧ j :={i if i ≤ jj if j ≤ i i ∨ j :=

{j if i ≤ ji if j ≤ i

• Now consider a partial order ≤ on the set X and a subset A ⊆ X.Then we define the set A∗ of minimal respectively the set A

∗ ofmaximal elements of A to be the following

A∗ := { a∗ ∈ A | ∀ a ∈ A : a ≤ a∗ =⇒ a = a∗ }A∗ := { a∗ ∈ A | ∀ a ∈ A : a∗ ≤ a =⇒ a = a∗ }

And an element a∗ ∈ A∗ is said to be a minimal element of A. Likewisea∗ ∈ A∗ is said to be a maximal element of A. Note that in generalit may happen that A has several minimal (or maximal) elements oreven none at all (e.g. N∗ = { 0 } and N∗ = ∅). For a linear orderminimal (and maximal) elements are unique however.

12 0 Prelude

• Finally ≤ is said to be a well-ordering on the set X, iff ≤ is a linearorder on X such that any non-empty subset A ⊆ X has a (alreadyunique) minimal element. Formally that is

∀ ∅ 6= A ⊆ X ∃ a∗ ∈ A such that ∀ a ∈ A : a∗ ≤ a

• Let X be any set, then the cardinality of X is defined to be the classof all sets that correspond bijectively to X. Formally that is

|X| := {Y | ∃ω : X → Y bijective }

Note that most textbooks on set theory define the cardinality to be acertain representative of our |X| here. However the exact definition isof no importance to us, what matters is comparing cardinalities: wedefine the following relation ≤ between cardinals:

|X| ≤ |Y | :⇐⇒ ∃ ι : X → Y injective⇐⇒ ∃π : Y → X surjective

|X| = |Y | :⇐⇒ ∃ω : X → Y bijective⇐⇒ |X| ≤ |Y | and |Y | ≤ |X|

Note that the first equivalence can be proved (as a standard exercise)using equivalence relations and a choice function (axiom of choice).The second equivalence is a rather non-trivial statement called theequivalence theorem of Bernstein. However these equivalences grantthat ≤ has the properties of a partial order, i.e. reflexivity, transitivityand anti-symmetry.

• Suppose x1, x2, . . . , xn are pairwise distinct elements. Then the setX := {x1, . . . , xn } clealy has n elements. Using cardialities this wouldbe expressed, as

| {x1, . . . , xn } | = |1 . . . n|

In this case 1 . . . n ←→ X : i 7→ xi could be used as a bijection. Yetthis is a bit cumbersome and hence we introduce another notation - ifx1, x2, . . . , xn are pairwise distinct (that is xi = xj =⇒ i = j) we let

# {x1, . . . , xn } := n

In particular #∅ = 0. Hence for any finite set X we have defined thenumber of elements #X. And for infinite sets we simply let

#X := ∞

• Thereby a set X is said to be finite iff there is some n ∈ N such thatX ←→ (1 . . . n), that is iff #X = n. And we usually denote the setof all finite subsets of X by

Ω(X) := {A ⊆ X | #A

Likewise X is said to be infinite, iff N can be embedded into X,formally N ↪→ X or in other words |N| ≤ |X|. Note that thereby weobtain a rather astounding equivalency of the statements

(a) X is infinite

(b) |X| = |X ×X|(c) |X| = |Ω(X)|

• If X and Y are finite, disjoint sets, then clearly #(X ∪ Y ) = (#X) +(#Y ). Likewise for any finite sets X and Y we have #(X × Y ) =(#X) · (#Y ), #P(X) = 2#X and #F(X,Y ) = (#Y )#X . We will usethese properties to extend the ordinary arithmetic of natural numbersto the arithmetic of cardinal numbers, by defining

|X|+ |Y | := |X t Y ||X| · |Y | := |X × Y |

2|X| := |P(X)||Y ||X| := |F(X,Y )|

for arbitrary sets X and Y . Note that t refers to the disjoint unionof X any Y , that is X t Y := ({ 1 } × X) ∪ ({ 2 } × Y ). This is notnecessary, if X and Y are disjoint, in this case we could have takenX ∪ Y . However in general we have to provide a way of seperating Xand Y and this nicely performed by X t Y . It is a non-surprising butalso non-trivial fact, that the power set of X is truly larger that X

|X| < |P(X)|

• (♦) We will introduce and use several different notions of substruc-tures and isomorphy. In order to avoid eventual misconceptions, weemphasize the kinds of structures regarded by applying subscripts tothe symbols ≤ and � of substructures and ∼= of isomorphy. E.g. wewill write R ≤r S to indicate, that R is a subring of S, a �i R toindicate that a is an ideal of R and R ∼=r S to indicate that R and Sare isomorphic as rings. Note that the latter is different from R ∼=m S(R and S are isomorphic as modules). And this can well make sense, ifR ≤r S is a subring, then S can be regarded as an R-module, as well.We will use the same subscripts for generated algebraic substructures,i.e. 〈• 〉r for rings, 〈• 〉i for ideals and 〈• 〉m for modules.

14 0 Prelude

Notation

A, B, C matrices, algebras and monoidsD, E, F fieldsG, H groups and monoidsI, J , K index setsL, M , N modulesP , Q subsets and substructuresR, S, T rings (all kinds of)U , V , W vectorspaces, multiplicatively closed setsX, Y , Z arbitrary sets

a, b, c elements of ringsd, e degree of polynomials, dimensione neutral element of a (group or) monoid

f , g, h functions, polynomials and elements of algebrasi, j, k, l integers and indexesm, n natural numbersp, q residue classesr, s elements of further ringss, t, u polynomial variablesu, v, w elements of vectorspacesx, y, z elements of groups and modules

α, β, γ multi-indexesι, κ (canonical) monomorphismsλ, µ eigenvaluesµ, ν valuation or Euclidean function%, σ (canonical) epimorphisms%, σ, τ permutationsϕ, ψ homomorphismsΦ, Ψ isomorphisms

Ω (fixed) finite set

a, b, c idealsu, v, w other idealsp, q prime idealsr, t other prime idealsm, n maximal idealsf, g, h fraction ideals

A, B, C (ordered) bases or families of setsM, N more (ordered) basesE , F (canonical = euclidean) bases

0.3 Notation and Symbols 15

0.4 A Primer on Sets

In this book we assume that the reader already is familiar with naive settheory, nevertheless we will present the most frequently used notions here.If you had troubles understanding the concepts in the previous section thiswill surely help, but we will not cover set theory in any depth.

It all began with Cantor’s notion of a set: A ”set” is a gathering togetherinto a whole of definite, distinct objects of our perception or of our thought.True to this concept Cantor wrote x ∈M if the object x belongs to the setM (he said is an element of M) and x 6∈ M if not. For example ”tuesday”is an element of the set of days in a week. Or the reader is an element ofthe set of all beings that are able to read English. Sets can even be infinite:A line is the set of all points that lie on it.

Though this may not be entirely satisfactory from a formal point of viewthis concept sustained mathematics for centuries. We will even refine thisconcept by presenting a list of axioms (that is statements, that we agreeupon as being true, without question) that completely describe what wemay safely do with sets. This list of axioms will be (ZFC) [an abbreviationfor Zermelo, Fraenkel plus the axiom of choice] that modern mathematicsalmost always uses.

This also is the way formal set theory took: Hereby it is not explainedwhat a set is. It just picks up a formal language containing the relationsymbol ∈ and deduces all the other theorems from (ZFC). It is therebycompletely defined, what we may do with sets, but not what a set is. Inthis context a set is just a variable symbol of the formal language speakingof sets.

Later in this book (when we come to category theory) we will also useclasses. Naively speaking a class is just an arbitrary large collection of sets.Thereby it may well happen, that a certain class is a set again (if it containsnot too many sets), but we have to separate these notions in order to avoidthe contradictions, a devious mind may come up with. Again we will presenta list (NBG) of axioms (which abbreviates Neumann, Bernays, Gödel) thatdescribe how to deal with classes.

From a formal point of view we have a two-sorted language - one sort forsets and the other sort for classes. Then all the other theorems are deducedfrom (ZFC) appended by (NBG). So again formal mathematics does notanswer what a class is but evades the necessity to do so. Note that we willnot say a word about naive or formal logic, but take this for granted. Ourmain focus are relations: partial orders, equivalency relations and functions.

(0.1) Definition:If X and Y are any sets, then a relation R between these sets is just asubset of the cartesian product of X and Y

R ⊆ X × Y

We say that the elements x ∈ X and y ∈ Y are related under R, if theordered pair (x, y) ∈ X × Y is contained in R. In this case we write xRy.

xRy :⇐⇒ (x, y) ∈ R

16 0 Prelude

(0.2) Remark: (viz. 467)The cartesian product X×Y is thought to be the set of all pairs (x, y) wherex ∈ X and y ∈ Y . Formally we could define (x, y) to be

(x, y) := {x, {x, y } }

Using this construction it can be proved (distinguishing 8 cases and usingthe axiom of foundation) that for any u, x ∈ X and any v, y ∈ Y we havethe desired property of ordered pairs

(u, v) = (x, y) ⇐⇒ u = x and v = y

(0.3) Definition:If X and Y are any sets again, then a partial function f is a relationbetween X and Y that is right unique in the following sense: For any x ∈ Xand any y1, y2 ∈ Y we have the implication

xfy1 and xfy2 =⇒ y1 = y2

In this case we will write f : X � Y and given x ∈ X and y ∈ Y we willoftenly rewrite the relation f in the following form

f : x 7→ y :⇐⇒ xfy

We will denote by f(x) the unique element y ∈ Y such that xfy (if thisexists). And we define the domain of f to be the set of all x ∈ X that areassigned some value y ∈ Y

dom(f) := {x ∈ X | ∃ y ∈ Y : xfy }

And a partial function is said to be a function, iff it is left total in thefollowing sense: For any x ∈ X there is some y ∈ Y such that xfy. Thus afunction f is a relation that satisfies

∀x ∈ X ∃ ! y ∈ Y : xfy

In other words that is dom(f) = X. Thereby a partial function f : X � Yalways is a function f : dom(f)→ Y . The set of all functions from X to Ywill be denoted by

F(X,Y ) = Y X := { f ⊆ X × Y | f is a function }

If now A ⊆ X and B ⊆ Y are any subsets of X and Y respectively thenwe define the image of A and preimage of B under f to be

f(A) := { f(x) | x ∈ A } ⊆ Yf−1(B) := {x ∈ X | f(x) ∈ B } ⊆ X

0.4 A Primer on Sets 17

(0.4) Remark:Intuitively speaking a function is a black box, that assigns every x ∈ X somevalue y = f(x) ∈ Y . To use a relation as the general framework just is aneat trick to pull functions back into the concept of sets. In fact this notionis so predominant that one oven introduces a symbol to get f back from f :The graph of f is defined to be

Γ(f) := { (x, y) ∈ X × Y | y = f(x) } = f

We will oftenly encounter the following situation: Given sets X, Y andsome x ∈ X we define some f(x). Then we need to prove that f : X → Ywith f : x 7→ f(x) is a well-defined function f : X → Y . This is to saywe have to check the following properties: (1) First of all for any x ∈ Xthe element f(x) needs to be an element of f(x) ∈ Y . And (2) if x1 andx2 ∈ X such that x1 = x2 then we need f(x1) = f(x2). This is the rightuniqueness of f . That is we have a partial function f :� Y . If we madeno restriction as to what x ∈ X we take, f already is a function from X to Y .

(0.5) Proposition: (viz. 468)Let X and Y be arbitrary sets, A ⊆ X and B ⊆ Y be subsets and considera function f : X → Y , then we get

A ⊆ f−1(f(A)

)f(X) ∩B = f

(f−1(B)

)X \ f−1(B) = f−1

(Y \B

)If now A = {Ai | i ∈ I } is an entire collection of subsets Ai ⊆ X andB = {Bj | j ∈ j } be a collection of subsets of Y , then we further get

f−1

⋃j∈J

Bj

= ⋃j∈J

f−1(Bj)

f−1

⋂j∈J

Bj

= ⋂j∈J

f−1(Bj)

f

(⋃i∈I

Ai

)=

⋃i∈I

f(Ai)

f

(⋂i∈I

Ai

)⊆

⋂i∈I

f(Ai)

(♦) Note that in the cases where we have an inclusion ⊆ only, then equalityneed not be true in general. However if f is injective, then in both cases wehave the equality of sets

A = f−1(f(A)

)f

(⋂i∈I

Ai

)=

⋂i∈I

f(Ai)

18 0 Prelude

(0.6) Definition:Let now X, Y and Z be sets, A ⊆ X be a subset and consider the functionsf : X → Y and g : Y → Z. First of all we define the restriction of f to Ato be the following function

f∣∣A

:= { (x, y) | x ∈ A, y = f(x) }

In particular the restriction f |A is a function of the form A → Y given byf |A : x 7→ f(x). And we also define the composition gf of g after f to bethe following function

gf := { (x, z) ∈ X × Z | ∃ y ∈ Y : y = f(x) and z = g(y) }

In particular gf is a function of the form X → Z that takes x ∈ X toy = f(x) ∈ Y and further on to z = g(y) ∈ Z. That is we can write

gf : X → Z : x 7→ g(f(x)

)

(0.7) Remark:Note that thereby the composition of maps is associative: That is for anychain of functions f : U → X, g : X → Y and h := Y → Z we get(hg)f = h(gf) as for any u ∈ U we have(

(hg)f)(u) = hg

(f(u)

)= h

(g(f(u)

))= h

(gf(u)

)=(h(gf)

)(u)

(0.8) Definition: (viz. 469)Let X and Y be arbitrary sets, f : X → Y be a function from X to Y .Then f is said to be injective, iff it satisfies any one of the following fourequivalent properties. And in this case we write f : X ↪→ Y .

(a) The function f admits a left-inverse that is there is some functiong : Y → X such that the composition satisfies

gf = 11X

(b) Speparate points of X will be assigned seperate points in Y : That isfor any u, x ∈ X we obtain the following implication

f(u) = f(x) =⇒ u = x

(c) For any subset A ⊆ X the preimage of the image of A is just A again

f−1(f(A)

)= A

(d) Every fiber of f contains at most one element: That is for any y ∈ Y

#f−1({ y }

)≤ 1

0.4 A Primer on Sets 19

And f : X → Y is said to be surjective, iff it satisfies any one of thefollowing four equivalent properties. And in this case we write f : X � Y .

(a) The function f admits a right-inverse that is there is some functiong : Y → X such that the composition satisfies

fg = 11 Y

(b) The function f reaches the entire set Y , that is the image of f is Y

f(X) = Y

(c) For any subset B ⊆ Y the image of the preimage of B is just B again

f(f−1(B)

)= B

(d) Every fiber of f contains at least one element: That is for any y ∈ Y

#f−1({ y }

)≥ 1

Finally f : X → Y is said to be bijective, iff it satisfies any one of thefollowing four equivalent properties. And in this case we write f : X ←→ Y .

(a) The function f is a one-to-one correspondence between the elementsof X and the elements of Y , that is for any y ∈ Y there is a uniquelydetermined x ∈ X such that y = f(x). Formally

∀ y ∈ Y ∃ ! x ∈ X : y = f(x)

(b) The function f has a uniquely defined inverse function f−1 : Y → Xwhose compositions satisfy

f−1f = 11X and ff−1 = 11 Y

(c) The function f admits both, a left-inverse function ` : Y → R and aright-inverse function r : Y → X that satisfy

`f = 11X and fr = 11 Y

(d) The function f is both: injective and bijective.

(0.9) Remark:For the time being f−1 is just a peculiar name for the inverse function off . It has the advantage of designating the function f it came from how-ever. This is useful as [due to the symmetry of property (a)] f is the inversefunction of f−1. Later we will denote the inverse element of some elementx ∈ G of a group G by x−1. This is consistent, as f−1 is precisely the inverseelement of f in the group of bijective functions from X to Y .

20 0 Prelude

Equivalence Relations

(0.10) Definition:Let X be a set and consider an arbitrary collection A = {Ai | i ∈ I } ofsubsets Ai ⊆ X. Then we say that A is a partition of X iff the Ai ∈ Aare non-empty, pairwise disjoint and X is the grand union of A. That is weget the following properties

(1) ∀Ai ∈ A : Ai 6= ∅

(2) ∀Ai, Aj ∈ A : Ai ∩Aj 6= ∅ =⇒ Ai = Aj(3) X =

⋃i∈I

Ai

And a relation of the form∼ ⊆ X×X is said to be an equivalence relationiff it is reflexive, symmetric and transitive. That is for any x, y and z ∈ Xwe find the following properties for ∼

(1) x ∼ x

(2) x ∼ y =⇒ y ∼ x

(3) x ∼ y, y ∼ z =⇒ x ∼ z

If ∼ is an equivalency relation on X and x ∈ X is any element, then wedefine the equivalency class of x to be the following subset of X

[x] := {u ∈ X | u ∼ x }

And the collection of all equivalency classes of ∼ will be denoted as (notethat the definition of this set may contain redundancies [u] = [x])

X/∼ := { [x] | x ∈ X }

(0.11) Lemma: (viz. 470)If X 6= ∅ is a non-empty set then there is a one-to-one correspondencebetween the partitions of X and the equivalency relations on X. That is ifwe denote the sets

part(X) := {A ⊆ P(X) | A is a partition }equi(X) := {∼ ⊆ X ×X | ∼ is an eqivalence relation }

then the following map from the set of equivalence relations on X to the setof partitions of X is well-defined and bijective

equi(X) ←→ part(X) : ∼ 7→ X/∼

Thereby the inverse function of this map is given by the following construc-tion: If A is a partition of X and u, x ∈ X we define the relation

u ∼A x :⇐⇒ ∃A ∈ A : {u, x } ⊆ A

0.4 A Primer on Sets 21

(0.12) Example:If X and Y are sets, f : X → Y is any function, then we obtain an equiva-lency relation ∼f of X by defining (for any u, y ∈ X)

u ∼f x :⇐⇒ f(u) = f(x)

It is clear that this relation is reflexive, symmetric and transitive (as theequality relation = satisfies all these properties). And for any x ∈ X theequivalency of x is precisely the fiber of f(x) ∈ Y , that is

[x] = f−1({ f(x) }

)⊆ X

(0.13) Remark:If ∼ is an equivalency relation on the set X, then the set X/ ∼ of equivalencyclasses is a partition of X. In particular any A ∈ X/ ∼ is non-empty andhence the axiom of choice guarantees, that there is a set R ⊆ X such thatwe have a one-to-one correspondence

R ←→ X/∼ : x 7→ [x]

Such a set R is called a system of representants of the relation ∼. Andwith such a system we have (without redundancies this time)

X/∼ = { [x] | x ∈ R }

Partial and Total Orders

(0.14) Definition:Let X be a set and ≤ ⊆ X ×X be a relation on X. Then we say that ≤ isa partial order on X, iff it is reflexive, transitive and antisymmetric. Thatis for any x, y and z ∈ X we find the the following properties for ≤

(1) x ≤ x

(2) x ∼ y, y ≤ z =⇒ x ∼ z

(3) x ∼ y, y ∼ x =⇒ x = y

A partial order ≤ is said to be a total order or alternatively linear order,iff it satisfies the following fourth property

∀x, y ∈ X : x ≤ y or y ≤ x

If ≤ is a partial [resp. total] order, then the pair (X,≤) is called a partiallyordered [resp.totally ordered] set. In the literature this is sometimesabbreviated as poset or toset respectively. In any case a subset N ⊆ X issaid to be a net in (X,≤) iff it satisfies

∀x, y ∈ N ∃ z ∈ N : x ≤ z and y ≤ z

22 0 Prelude

And a subset C ⊆ X said to be a chain in (X,≤) iff is a totally orderedset under the order ≤ inherited from X. That is iff it satisfies

∀x, y ∈ C : x ≤ y or y ≤ x

Let us denote the set of all nets, resp. of all chains resp. of all finite chainsin (X,≤) by N (X,≤), C(X,≤) and C0(X,≤), formally

N (X,≤) := {N ⊆ X | C net }C(X,≤) := {C ⊆ X | C chain }C0(X,≤) := {C ⊆ X | C chain, #C

(0.15) Remark:

(i) Let X be any set, recall the definition of the power set to be the setof all subsets of X, that is

P(X) = {A | A ⊆ X }

A special case is the power set of the empty set, which is P(∅) = { ∅ }.And e.g. for X = { 1, 2, 3 } the power set is given to be

P(X) = {∅, { 1 } , { 2 } , { 3 } , { 1, 2 } , { 1, 3 } , { 2, 3 } , X}

In general P(X) contains 2|X| elements. Thereby (P(X), ⊆ ) is apartially ordered set for any set X, however ⊆ is not a total or-der. This can already bee seen in the example here: The subsets{ 1, 2 } and { 1, 3 } are not comparable under ⊆ , as we neither have{ 1, 2 } ⊆ { 1, 3 } nor { 1, 3 } ⊆ { 1, 2 }.

(ii) If (X,≤) is a partially [resp. totally] ordered set and A ⊆ X is asubset, then the restriction ≤ ∩A×A ⊆ A×A is a partial [resp. total]order on A. By abuse of notation we write ≤ for the restricted orderas well, that is (A,≤) is a partially [resp. totally] ordered set again.

(iii) In particular any collection A ⊆ P(X) of subsets of an arbitrary setX is partially ordered under ⊆ , that is

(A, ⊆ ) is a partially ordered set

(iv) Let us continue with the example X = { 1, 2, 3 } in (i) and regard thecollection A := P(X) \ { ∅, X } of subsets of X, explicitly

A = {{ 1 } , { 2 } , { 3 } , { 1, 2 } , { 1, 3 } , { 2, 3 }}

Then A has 3 maximal elements, namely { 1, 2 }, { 1, 3 } and { 2, 3 }.That is the set of maximal elements of A is given to be

A∗ = {{ 1, 2 } , { 1, 3 } , { 2, 3 }}

(v) (�) Probably the most important example of all are the natural num-bers N. They admit an ordering m ≤ n :⇐⇒ ∃ k ∈ N : m + k = nthat even is a well-ordering. That is any non-empty subset N ⊆ Nhas a least element. The reader is asked to consult any textbook on”numbers” for a construction of N and a proof of this theorem.

24 0 Prelude

(0.16) Proposition: (viz. 472)If (X,≤) is a partially ordered set and F ⊆ X is a non-empty, finite subsetof X, then F contains at least one maximal and one minimal element

F ∗ 6= ∅ and F∗ 6= ∅

If (X,≤) is a totally ordered set and F ⊆ X is a non-empty, finite subsetof X, then F has both a greatest and a least element

∃ ! g, ` ∈ F : max(F ) = g and min(F ) = `

(0.17) Proposition: (viz. 472)If (X,≤) is a partially ordered set and for any i ∈ I we consider a chainCi ∈ C(X,≤) (where I 6= ∅ is an arbitrary index set), then their intersectionis a chain in X again ⋂

i∈ICi ∈ C(X,≤)

And if {Ci | i ∈ I } is a net under inclusion [i.e. if it is a net in the partiallyordered set (P(X), ⊆ )] then their union is a chain in X, again⋃

i∈ICi ∈ C(X,≤)

If (X,≤) and (Y,≤) are partially ordered set then a funcion f : X → Yis said to be mono-tonous, iff x ≤ y implies f(x) ≤ f(y). In other wordsf is order-preserving. If conversely x ≤ y implies f(x) ≥ f(y), then f issaid to be anti-tonous or order-reversing. The following situation is rathercommonplace - we will encounter it several times:

(0.18) Definition: (viz. 472)Let (X,≤) and (Y,≤) be two partially ordered sets and let f : X → Y andg : Y → X be two order-reversing functions, that is for any x, x′ ∈ X andany y, y′ ∈ Y we have the implications

x ≤ x′ =⇒ f(x′) ≤ f(x)y ≤ y′ =⇒ g(y′) ≤ g(y)

Then the following three statements are equivalent. And in this case we saythat (f, g) is a Galois connection between (X,≤) and (Y,≤).

(a) ∀x ∈ X, ∀ y ∈ Y : x ≤ g(x) ⇐⇒ y ≤ f(x)

(b) ∀x ∈ X, ∀ y ∈ Y : x ≤ gf(x) and y ≤ fg(y)

(c) For any x ∈ X the element f(x) ∈ Y is the greatest element of Y , thatsatisfies x ≤ g(y) and likewise for g. That is (for any x ∈ X and anyy ∈ Y ) we have

f(x) = max{ y ∈ Y | x ≤ g(y) }g(y) = max{x ∈ X | y ≤ f(x) }

0.4 A Primer on Sets 25

And in this case these maps satisfy fgf = f and gfg = g. In other wordsthe functions f and g give rise to a one-to-one correspondence between theirrespective image sets. That is the following map is bijective:

g(Y ) ←→ f(X)x 7→ f(x)

g(y) ←[ y

(0.19) Example:The properties fgf = f and gfg = g does not suffice for a Galois connectionhowever. As an example take X := { 1, 2, 3 } and Y := { 1, 2 } under thenatural order of N. Then define f by f(1) := 2, f(2) := 1 and f(3) := 1.Likewise g(1) := 2 and g(2) := 1. Then straightforward to check that weget the one-to-one correspondence { 1, 2 } ←→ { 1, 2 } but gf(3) = 2 doesnot satisfy 3 ≤ gf(3), that is (b) is not satisfied.

Chain Conditions

(0.20) Definition: (viz. 473)Let (X,≤) be a partially ordered set, then we say that (X,≤) satisfies theascending chain condition (which we will always abbreviate by ACC),iff it satisfies one of the following two equivalent properties

(a) Any non-empty A 6= ∅ subset A ⊆ X of X has a maximal element.That is there is some a∗ ∈ A such that

∀ a ∈ A : a ≤ a∗ =⇒ a = a∗

(b) Any ascending chain within X stabilizes, that is for any (xk) ⊆ X(where k ∈ N) such that x0 ≤ x1 ≤ x2 ≤ . . . we have

∃ s ∈ N ∀ i ∈ N : xs+i = xs

In complete analogy we we say that (X,≤) satisfies the descending chaincondition (which we will always abbreviate by DCC), iff it satisfies one ofthe following two equivalent properties

(a) Any non-empty A 6= ∅ subset A ⊆ X of X has a minimal element.That is there is some a∗ ∈ A such that

∀ a ∈ A : a∗ ≤ a =⇒ a = a∗

(b) Any descending chain within X stabilizes, that is for any (xk) ⊆ X(where k ∈ N) such that x0 ≥ x1 ≥ x2 ≥ . . . we have

∃ s ∈ N ∀ i ∈ N : xs+i = xs

26 0 Prelude

(0.21) Lemma: (viz. 474)Let (X,≤) be a partially ordered set and recall that C(X,≤) denotes the setof all chains, C0(X,≤) denotes the set of finite chains of (X,≤). Then thefollowing three statements are equivalent

(a) (X,≤) satisfies both ACC and DCC

(b) (C0(X,≤), ⊆ ) satisfies ACC

(c) Every chain in (X,≤) is finite, that is the have the equality of the sets

C(X,≤) = C0(X,≤)

And in this case any chain in (X,≤) can be refined to a maximal finite chain,that is: If C ⊆ X is a chain in (X,≤), then there is some C∗ ∈ C0(X,≤)such that C ⊆ C∗ and

∀D ∈ C(X,≤) : C∗ ⊆ D =⇒ C∗ = D

The Lemma of Zorn

(0.22) Theorem: (viz. 475)Hausdorff’s Maximum Principle: Let (X,≤) be a partially ordered set, thenthere is a chain M ⊆ X in X, that is a maximal element under inclusion:

∀C ∈ C(X,≤) : M ⊆ C =⇒ C = M

(0.23) Corollary: (viz. 478)Lemma of Zorn: Let (Z,≤) be a partially ordered set, such that the followingtwo properties are satisfied (1) Z 6= ∅ is non-empty and (2) every chain inZ has an upper bound, that is: for any C ⊆ Z there is some u ∈ Z suchthat for any c ∈ C we have c ≤ u. Then Z already has a maximal element,that is: for some m ∈ Z any z ∈ Z satisfies m ≤ z =⇒ m = z. Formally:

if ∀C ∈ C(Z,≤) ∃u ∈ Z : ∀ c ∈ C : c ≤ u

then ∃m ∈ Z ∀ z ∈ Z : m ≤ z =⇒ z = m

(0.24) Remark:We will mostly use the following special case of Zorn’s Lemma: Let X beany set and Z ⊆ P(X) be a non-empty Z 6= ∅ family of subsets of X suchthat with any chain in C ⊆ Z the union

⋃C ∈ Z is contained in Z as well,

then Z has a maximal element: That is there is some M ∈ Z such that forany Z ∈ Z we have M ⊆ Z =⇒ Z = M . Formally again

if ∀ C ∈ C(Z, ⊆ ) :⋃C ∈ Z

then ∃M ∈ Z ∀Z ∈ Z : M ⊆ Z =⇒ Z = M

0.4 A Primer on Sets 27

Prob (Z, ⊆ ) is a partially ordered set, such that Z 6= ∅ if now C ⊆ Z is achain, then we let U :=

⋃C. By assumption we have U ∈ Z and C ⊆ U

for any C ∈ C is clear. Then the lemma of Zorn yields M ∈ Z.

We will use the lemma of Zorn to prove the existence of maximal ideals andmodules with all kinds of properties. It can also be used to construct partialfunctions with maximal domain and so on. It is the single most powerfultool at or disposal! But though it is easy to use and very versatile (due toits generality) it only grants the existence of a maximal element. It does nottell us how to find or construct it.

(0.25) Theorem: (viz. 478)Well-ordering Theorem: Let X be any set, then there is a total order ≤ on Xthat even satisfies the following well-ordering property: For any non-emptysubset A ⊆ X, A 6= ∅ there is a least element m ∈ A, that is

∃m ∈ A : ∀ a ∈ A : m ≤ a

(0.26) Remark:These are the three basic theorems of set theory. And if we take a closelook at the proofs [the reader is not required to do so, in order do be agood mathematician, but an understanding of the content of the theoremsis required] we see that Hausdorff’s maximum principle is derived from theaxiom of choice, Zorn’s lemma is derived from the maximum principle andthe well-ordering theorem is derived from Zorn’s lemma. We just wantto sketch the idea that we can go full circle: The axiom of choice can bederived from the well-ordering theorem easily: Given an arbitrary collectionA = {Ai | i ∈ I } collection of non-empty sets Ai 6= ∅ we can construct achoice function in the following way: Let A :=

⋃iAi andX :=

⋃i { i }×Ai ⊆

P(I ×A) be the disjoint union of the Ai. Now install a well-ordering ≤ onX, then we may define a choice function f : A → A by letting f(Ai) := xiwhere xi is the minimum of the set { i } × Ai ⊆ X under ≤, formally(i, xi) = min({ i } × Ai). Therefore these three properties are equivalent onthe basis of the other axioms of set theory. It is a different story, that theyeven are independent of the other axioms of set theory, that is they canneither be proved nor falsified. But this result requires much more effort informal logic.

28 0 Prelude

0.5 A Primer on Classes

Let us begin this section with a list of the different axioms of set theory incommented form, hoping that the interested reader can thereby get an ideahow Cantor’s notion can be formalized. At times this will require to givetwo formulations: one of naive set theory and the other in (almost) formallogic. Later on we will sketch how to extend these axioms to handle classes:

(0.27) Remark: (ZFC)

(1) Axiom of Extensionality: Using Cantor’s notion of a set it is clearthat any sets X and Y are equal, if and only if they contain the sameelements. That is why we can prove X = Y by proving X ⊆ Yand Y ⊆ X separately. The first axiom formalizes this property, itsformula reads

∀X ∀Y : X = Y ⇐⇒ ∀ a(a ∈ X ⇐⇒ a ∈ Y

)(2) Axiom of Union: Given any collection of sets A = {Ai | i ∈ I } we

often use the grand union A =⋃iAi of these sets. Then a is an

element of A if and only if a ∈ Ai for some i ∈ I. The following axiomguarantees that the set A truly exists

∀A∃A : ∀ a(a ∈ A ⇐⇒ ∃Ai : (Ai ∈ A and a ∈ Ai)

)(3) Axiom of Power Set: For any set X there is a set containing all the

subsets of X. This is called the power set P(X) = {A | A ⊆ X } ofX. The following axiom guarantees that the power set truly exists

∀X ∃P : ∀A(A ∈ P ⇐⇒ A ⊆ X

)thereby the inclusion of sets A ⊆ X is an abbreviation for anotherformula: A ⊆ X is defined, as ∀ a (a ∈ A =⇒ a ∈ X).

(4) Scheme of the Axioms of Separation: Given a set X and a predicate ϕon X (this is a function ϕ : X → { true, false } of X we would like topick up the set {x ∈ X | ϕ(x) } that consists of all the elements x ∈ Xsuch that ϕ(x) = true. For Cantor the existence of this set is evident,in formal set theory the purpose of the list of axioms is to pin downwhat we take for granted. Thus for every predicate ϕ we allow

∀X ∃A : ∀x(x ∈ A ⇐⇒

(x ∈ X and ϕ

))So what is a predicate in the context of this scheme of axioms? Ofcourse ϕ is a formula in the formal language of sets. And this formulahas to contain x as a free variable, that is x is a variable symbol that isnot bound by a quantifier ∀ or ∃ already (a notion that is best definedby recursion on the setup of the formula). For this scheme we requirethat x is the one and only free variable symbol of ϕ.

0.5 A Primer on Classes 29

(5) Axiom of Infinity: It is possible to construct the natural numbers N interms of set theory: We start with 0 := ∅ and iteratively define n+ 1to be the set n∪{n }. The set containing 0, 1 = 0+1, 2 = 1+1 and soon is N and it can be shown that any other set N that is generated bya succession function N → N : n 7→ n+ from a starting element has anorder-preserving bijection to N. In this sense the natural numbers areunique up to isomorphy. The natural numbers are such a basic notionthat nobody would doubt them, but in formal set theory we have toguarantee their existence by an axiom:

∃N(∅ ∈ N and ∀n

(n ∈ N =⇒ n+ ∈ N

))In order to break this formula down into the language of formal settheory we have to make several replacements: The empty set ∅ isdefined by E = ∅ ⇐⇒ ∀x(x ∈ ∅ ⇐⇒ x 6= x) Thus instead of ∅ ∈ Nwe would have to write ∃E(E = ∅ and E ∈ N). The existence of ∅ isguaranteed by (4), just take the predicate ϕ = (x 6= x). So it remainsto define n+ = n∪{n }: Clearly a ∈ n+ iff a ∈ n or a = n. Thus we lets = n+ ⇐⇒ ∀ a(a ∈ n or a = n). Therefore n+ needs to be replacesby ∃ s (s = n+ and s ∈ N). The resulting formula is far too long to becaptured in one line and hence we will abstain from providing it here.

(6) Scheme of the Axioms of Replacement: If f : X � Y is a partialfunction and A ⊆ X is a subset of X, then we would like to map A(that is the part of A inside the domain of f) to Y . If a ∈ A is mappedunder f then the element b = f(a) ∈ Y is uniquely determined. There-fore we will get a subset B = f(A) = { f(a) | a ∈ A ∩ dom(f) } ⊆ Y .Let us make the existence of f(A) an axiom:

∀ f(f : X � Y =⇒ ∀A∃B ∀ b

(b ∈ B ⇐⇒ b ∈ f(A)

))where b ∈ f(A) :⇐⇒ ∃ a

(a ∈ A and b = f(a)

)It requires some effort to formulate this in the language of sets. Firstof all we have to replace the partial function f : Let ϕ be a formula inthe language of sets having the the free variables x and y [refer to (4)for this]. If now a and b are any variable symbols, then ϕ(x : a, y : b)denotes the formula where every free occurance of x is replaced by aand any free occurance of y is replaced by b. We now recall how weintroduced functions as sets: we defined b = f(a) by afb. Thus wereplace b = f(a) by ϕ(x : a, y : b). Consequently f : X � Y has tobe replaced by

∀ a ∀ b1 ∀ b2((ϕ(x : a, y : b1) and ϕ(x : a, y : b2)

)=⇒ b1 = b2

)It remains to replace the quantifier ∀ f , as we replaced f by a formulaϕ. Hence we take this formula as an axiom for every formula ϕ thathas the free variables x, y and s1, . . . , sn (n ∈ N) but no others. Thenwe replace ∀ f by ∀ s1 . . . ∀ sn and are finally done.

(7) Axiom of Foundation: Cantor would say that no set can satisfyX ∈ X,by the following reasoning: In order to build up the set X we need toknow all its elements. So X cannot be on of its own elements, since by

30 0 Prelude

the time we think of the set X the element X is incomplete. For thesame reason there can be no circle of sets X1 ∈ X2, X2 ∈ X3 and soon Xn−1 ∈ Xn with Xn ∈ X1 again. In formal set theory one usuallytakes a little stronger property to be an axiom: Every non-empty setX contains an element d, that is disjoint to X, as a formula:

∀X(X 6= ∅ =⇒ ∃ d

(d ∈ X and d ∩X = ∅

))In the formal language we only have to make little adjustments: X 6= ∅needs to be reformulated as ∃uu ∈ X and d∩X = ∅ has to be replacedby ∀ v(v ∈ x and v ∈ X =⇒ v 6= v). It is easy to deduct the non-existence of circles from this axiom (see below). And using the axiomof choice we can also deduct this formula from the non-existence ofcircles of sets.

Prob Suppose we had a circle of sets x1, . . . , xn such that xk ∈ xk+1and xn ∈ x1. Then we consider X := {x1, . . . , xn } and choose somed ∈ X such that d ∩X = ∅. By construction we have d = xk for somek ∈ 1 . . . n. Thus xk−1 ∈ d or xn ∈ d for k = 1. That is xi ∈ d forsome i ∈ 1 . . . n. But then xi ∈ d ∩X in contradiction to d ∩X = ∅.

(8) Axiom of Choice: This has often been stated: Given an arbitrarycollection of sets A = {Ai | i ∈ I } such that Ai 6= ∅ is non-empty (forany i ∈ I) there is a set A = { ai | i ∈ I } such that for any i ∈ I theelement ai is taken from ai ∈ Ai. We say that A is a choice from A.In fact it suffices to claim a little less: If P is a partition of some setX then there is a set A ⊆ X such that for any a ∈ A there is exactlyone P ∈ P such that a ∈ P . (see below). We will now formalize thisproperty as an axiom:

∀P(P partition =⇒ ∃A : A choice from P

)Of course ”P partition” abbreviates a formula in the language of sets,namely ∅ 6∈ P and (∀P ∀Q . . . ) where . . . is given to be the following

P ∈ P and Q ∈ P and ∃x(x ∈ P and x ∈ Q

)=⇒ P = Q

Likewise ”A choice from P” abbreviates another formula, this time

∀P(P ∈ P =⇒ ∃ ! a

(a ∈ A and a ∈ P

))Prob Let us define Pi := { i }×Ai and X :=

⋃Pi Then it is clear that

P := {Pi | i ∈ I } is a partition of X. Thus there is some P ⊆ Xsuch that for any p ∈ P there is a unique Pi ∈ P with p ∈ Pi. Byconstruction p is of the form p = (i, a) for some a ∈ Ai. Now defineπ : X →

⋃iAi by (i, a) 7→ a. Then A := π(P ) is a choice from A.

0.5 A Primer on Classes 31

Intuitively speaking classes are collections of sets that are just too large tobe a set themselves. E.g. if we denote U the collection of all sets (some-times also called the universe) and would think that this could be a set,then we would have U ∈ U which we shun. As we wish to address suchconstructions, we need a new term for these, in order to avoid contradic-tions. This new term is class, that’s all. In fact there are classes, that canwell be sets, these classes are called small, but the interesting classes are not.

Warning: The rest of this section will advance set-theory by introducingclasses, as a second, fundamental concept. This will definitely not be neededprior to chapter 12 where we proceed on to categorial algebra. And eventhere a naive understanding of classed is all that is required. Do not ventureon with this section, unless you have a solid working knowledge of set theoryand feel comfortable with the axioms of ZFC. And if you are interested instandard algebra, skip this section entirely. You will not miss a single thing.

From a formal point of view we extend the language of sets (with only onesort of variable symbols, which we interpret as sets) to a language of sets andclasses (with two sorts of variable symbols, one for sets and one for classes).Most textbooks try to avoid this by introducing two predicate symbols onesaying I am a set and the other saying I am a class, but accepting the twosorts is the natural way to go. Thus in this two-sorted language we will haveseparate quantifiers for the different sorts:

universal quantifier existential quantifier

sets ∀s ∃sclasses ∀c ∃c

For example the formula ∀sX ∃c C : X = C says: For every set X there isa class C that is equal to X. This will even be the first of our axioms forclasses. So here comes the rest: A commented list of (NBG)

32 0 Prelude

Part I

The Truth

Chapter 1

Groups and Rings

1.1 Defining Groups

The most familiar (and hence easiest to understand) objects of algebra arerings. And of these the easiest example are the integers Z. On these thereare two operations: an addition + and a multiplication ·. Both of theseoperations have familiar properties (associativity for example). So we firststudy objects with a single operation ◦ only - monoids and groups - as theseare a unifying concept for both addition and multiplication. However ouraim solely lies in preparing the concepts of rings and modules, so the readeris asked to venture lightly over problems within this section until he hasreached the later sections of this chapter.

(1.1) Definition:Let G 6= ∅ be any non-empty set and ◦ a binary operation on G, i.e. ◦ is amapping of the form ◦ : G × G → G : (x, y) 7→ xy. Then the ordered pair(G, ◦) is said to be a monoid iff it satisfies the following properties

(A) ∀x, y, z ∈ G : x(yz) = (xy)z

(N) ∃ e ∈ G ∀x ∈ G : xe = x = ex

Note that this element e ∈ G whose existence is required in (N) then alreadyis uniquely determined (see below). It is said to be the neutral elementof G. And therefore we may define: a monoid (G, ◦) is said to be a groupiff any element x ∈ G has an inverse element y ∈ G, that is iff

(I) ∀x ∈ G ∃ y ∈ G : xy = e = yx

Note that in this case the inverse element y of x is uniquely determined (seebelow) and we hence write x−1 := y. Finally a monoid (or group) (G, ◦) issaid to be commutative iff it satisfies the property

(C) ∀x, y ∈ G : xy = yx

(1.2) Remark:

• We have to append some remarks here: first of all we have employed afunction ◦ : G×G→ G. The image ◦(x, y) of the pair (x, y) ∈ G×Ghas been written in an unfamiliar way however

xy := ◦(x, y)

34

If you have never seen this notation before it may be somewhat start-ling, in this case we would like to reassure you, that this actually isnothing new - just have a look at the examples further below. Yet thisnotation has the advantage of restricting itself to the essential. If wehad stuck to the classical notation such terms would be by far moreobfuscated. E.g. let us rewrite property (A) in classical terms

◦(x, ◦(y, z)

)= ◦

(◦ (x, y), z

)• It is easy to see that the neutral element e of a monoid (G, ◦) is

uniquely determined: suppose that another element f ∈ G would sat-isfy ∀x ∈ G : xf = x = fx, then we would have ef = e by lettingx = e. But as e is a neutral element we have ef = f by (N) appliedwith x = f . And hence e = f are equal, i.e. e is uniquely determined.Hence in the following we will reserve the letter e for the neutral el-ement of the monoid regarded without specifically mentioning it. Incase we apply several monoids at once, we will name the respectiveneutral elements explictly.

• Property (A) is a very important one and hence it has a name of itsown: associativity. A pair (G, ◦) that satisfies (A) only also is calleda groupoid. We will rarely employ these objects however.

• Suppose (G, ◦) is a monoid with the (uniquely determined) neutralelement e ∈ G. And suppose x ∈ G is some element of G, that hasan inverse element. Then this inverse is uniquely determined: supposeboth y and z ∈ G satisfy xy = e = yx and xz = e = zx. Then theasociativity yields y = z, as we may compute

y = ye = y(xz) = (yx)z = ez = z

• Another important consequence of the associativity is the following:consider finitely many elements x1, . . . , xn ∈ G (where (G, ◦) is agroupoid at least). Then any application of parentheses to the productx1x2 . . . xn produces the same element of G. As an example consider

x1(x2(x3x4)) = x1((x2x3)x4) = (x1(x2x3))x4 = ((x1x2)x3)x4

In every step of these equalities we have only used the associativitylaw (A). The remarkable fact is that any product of any n elements(here n = 4) only depends on the order of the elements, not on thebracketing. Hence it is costumary to omit the bracketing altogether

x1x2 . . . xn :=(

(x1x2) . . .)xn ∈ G

And any other assignment of pairs (without changing the order) tothe elements xi would yield the same element as x1x2 . . . xn. A formalversion of this statement and its proof are given in chapter 17 of thisbook. The proof clearly will be done by induction on n ≥ 3, thefoundation of the induction precisely is the law of associativity.

1.1 Defining Groups 35

• Suppose (G, ◦) is any groupoid, x ∈ G is an element and 1 ≤ k ∈ N,then we abbreviate the k-nary product of x by xk, i.e. we let

xk := xx . . . x (k − times)

If (G, ◦) even is a monoid (with neutral element e) it is customary todefine x0 := e. Thus in this case xk ∈ G is defined for all k ∈ N. Nowsuppose that x even is invertible (e.g. if (G, ◦) is a group), then wemay even define xk ∈ G for any k ∈ Z. Suppose 1 ≤ k ∈ N, then

x−k :=(x−1

)k• In a commutative groupoid (G, ◦) we may even change the order in

which the elements x1, . . . , xn ∈ G are multiplied. I.e. if we are givena bijective map σ : 1 . . . n ←→ 1 . . . n on the indexes 1 . . . n we get

xσ(1)xσ(2) . . . xσ(n) = x1x2 . . . xn

The reason behind this is the following: any permutation σ can bedecomposed into a series of transpositions (this is intuitively clear:we can generate any order of n objects by repeatedly interchangingtwo of these objects). In fact any transposition can be realized byinterchanging adjacent objects only. But any transposition of adjacentelements is allowed by property (C). A formal proof of this reasoningwill be presented in chapter 17 again.

(1.3) Example:

• The most familiar example of a (commutative) monoid are the naturalnumbers under addition: (N,+). Here the neutral element is given tobe e = 0. However 1 ∈ N has no inverse element (as for any a ∈ N wehave a+ 1 > 0) and hence (N,+) is no group.

• The integers however are a (commutative) group (Z,+) under addi-tion. The neutral element is zero again e = 0 and the inverse elementof a ∈ Z is −a. Thus N is contained in a group N ⊆ Z.

• Next we regard the non-zero rationals Q∗ := { a/b | 0 6= a, b ∈ Z }.These form a group under multiplication (Q∗, ·). The neutral elementis given to be 1 = 1/1 and the inverse of a/b ∈ Q∗ is b/a.

• Consider any non-empty set X 6= ∅. Then the set of maps from Xto X, which we denote by F(X) := {σ | σ : X → X }, becomes amonoid under the composition of maps (F(X), ◦) (as the compositionof functions is associative). The neutral element is the identity mape = 11X which is given to be 11X : x 7→ x.

• Consider any non-empty set X 6= ∅ again. Then the set of bijectivemaps SX := {σ : X → X | σ bijective } ⊆ F(X) on X even becomesa group under the composition of maps (SX , ◦). The neutral elementis the identity map e = 11X again and the inverse of σ ∈ SX is theinverse function σ−1. This will be continued in the next section.

36 1 Groups and Rings

• (♦) A special case of the above is the set of invertible (n×n)-matricesgln(E) := {A ∈ matn(E) | det(A) 6= 0 } over a field (E,+, ·). This isa group (gln(E), ·) under the multiplication · of matrices.

(1.4) Remark:Consider a finite groupoid (G, ◦), that is the set G = {x1, . . . , xn } is finite.Then the composition ◦ can be given by a table of the following form

◦ x1 x2 . . . xnx1 x1x1 x1x2 . . . x1xnx2 x2x1 x2x2 . . . x2xn...

......

...xn xnx1 xnx2 . . . xnxn

Such a table is also known as the Cayley diagram of G. As an exampleconsider a set with 4 elements K := { e, x, y, z }, then the following diagramdetermines a group structure ◦ on K ((K, ◦) is called the Klein 4-group).

◦ e x y ze e x y zx x e z yy y z e xz z y x e

(1.5) Proposition: (viz. 482)Let (G, ◦) be any group (with neutral element e), x, y ∈ G be any twoelements of G and k, l ∈ Z. Then we obtain the following identities

e−1 = e(x−1

)−1= x(

xy)−1

= y−1x−1

xkxl = xk+l(xk)l

= xkl

In particular the inversion i : G ←→ G : x 7→ x−1 of elements is a self-inverse, bijective mapping i = i−1 on the group G. If now xy = yx docommute then for any k ∈ Z we also obtain

xy = yx =⇒ (xy)k = xkyk

1.1 Defining Groups 37

(1.6) Definition:If (G, ◦) and (H, ◦) are any two groups, whose neutral elements are e ∈ G andf ∈ H respectively, then a map ϕ : G→ H is said to be a homomophismof groups or group-homomophism, iff it satisfies

∀x, y ∈ G we get ϕ(xy) = ϕ(x)ϕ(y)

And in this case ϕ already links the neutral elements and maps inverseelements to the inverse, that is it satisfies ϕ(e) = f and

∀x ∈ G we get ϕ(x−1) = ϕ(x)−1

Let us introduce the set of all group-homomophisms from G to H to becalled

ghom(G,H) := {ϕ : G→ H | ∀x, y ∈ G : ϕ(xy) = ϕ(x)ϕ(y) }

Prob clearly ϕ(e) = ϕ(ee) = ϕ(e)ϕ(e) but as H is a group we may multiplyby ϕ(e)−1 to find f = ϕ(e). Therefore f = ϕ(e) = ϕ(xx−1) = ϕ(x)ϕ(x−1).Likewise we get f = ϕ(x−1)ϕ(x) and hence ϕ(x−1) = ϕ(x)−1.

(1.7) Proposition: (viz. 484)Let (G, ◦) be a group with neutral element e ∈ G, then a subset P ⊆ G issaid to be a subgroup of G (written as P ≤g G) iff it satisfies

e ∈ Px, y ∈ P =⇒ xy ∈ Px ∈ P =⇒ x−1 ∈ P

In other words P ⊆ G is a subgroup of G iff it is a group (P, ◦) under theoperation ◦ inherited from G. And in this case we obtain an equivalencerelation on G by letting (for any x, y ∈ G)

x ∼ y :⇐⇒ y−1x ∈ P

And for any x ∈ G the equivalence class of x is thereby given to be thecoset xP := [x] = {xp | p ∈ P }. We thereby define the index [G : P ] of Pin G to be the following cardinal number

G/P :=

G/∼

[G : P ] :=∣∣∣G/P ∣∣∣

And thereby we finally obtain the following identity of cardinals which iscalled the Theorem of Lagrange (and which means G ←→ (G/P )× P )

|G| = [G : P ] · |P |

38 1 Groups and Rings

(1.8) Proposition: (viz. 484)Let (G, ◦) be any monoid with neutral element e and X ⊆ G be any subsetof G. Then the intersection of all submonoids of G that contain X is asubmonoid of G again

〈X 〉o :=⋂{P ≤o G | X ⊆ P } ≤o G

We call 〈X 〉o the monoid generated by X ⊆ G. And letting Xe := X∪{ e }we can even give an explicit description of 〈X 〉o to be the following set

〈X 〉o = {x1x2 . . . xn | n ∈ N, x1, . . . , xn ∈ Xe }

If G even is a group, then we can regard the intersection of all subgroups ofG that contain X and this is a subgroup of X again

〈X 〉g :=⋂{P ≤o G | X ⊆ P } ≤g G

Likewise ee call 〈X 〉g the group generated by X ⊆ G. And letting X± :=X ∪ { e } ∪

{x−1 | x ∈ X

}we can even give an explicit description of 〈X 〉g

to be the following set

〈X 〉g = {x1x2 . . . xn | n ∈ N, x1, . . . , xn ∈ X± }

(1.9) Proposition: (viz. 485)If (G, ◦) is a group and x ∈ G is any element of G, then the above propositiontells us that the subgroup generated by x is given to be

〈x 〉g ={xk | k ∈ Z

}If now G is finite, that is n := #G ∈ N, then 〈x 〉g (being a subset) is finitetoo. In particular we may define the order of x to be number of elementsof 〈x 〉g. And thereby we find

ord(x) := #〈x 〉g = min{ 1 ≤ k ∈ N | xk = e }

That is k = ord(x) is the minimal number 1 ≤ k ∈ N such that xk = e.And the theorem of Lagrange even tells us that k divides n, that is

ord(x) | #G

1.1 Defining Groups 39

1.2 Permutations

In this section we will study the most important and most general exampleof groups: permutations. Permutation groups will not become importantuntil we reach determinants in chapter 4. So a novice might well skip theentire section and return to it only later. In fact we even recommend thisapproach, as we deem the basics of ring theory to be easier than the basicsof group theory. Also we will use some notation (namely products and thefaculty) that will only be introduced in sections 1.3 and 1.5 respectively.

Later in section 10.2 we will introduce group actions, that are a gener-alization of the permutation action. Also permutations are used to studyconjugation in a group and they are occasionally useful in other areas aswell. In fact they will be applied in Galois, representation and invarianttheory.

(1.10) Definition: (viz. 486)IfX 6= ∅ is any non-empty set then we define the group SX of permutationsof X to be the set of all bijective maps on X, that is

SX := {σ : X → X | σ is bijective }

Thereby SX truly becomes a group (SX , ◦) under usual the composition ◦of functions. An in case of X = 1 . . . n we will write

Sn := {σ : (1 . . . n)→ (1 . . . n) | σ is bijective }

(1.11) Remark:If ∅ 6= X ⊆ Y is a non-empty subset of some set Y , then it is clear that thepermutations on X can be regarded as a subset of the permutations on Y .To be precise we get the following monomorphism of groups:

SX ↪→ SY : π 7→ π

where π : Y → Y is defined by π(x) := π(x) for any x ∈ X and π(y) := yfor any y ∈ Y \ X. The image if this monomorphism is just the set of allσ ∈ SY that fix all the elements of Y \X, formally again:

SX ←→ {σ ∈ SY | ∀ y ∈ Y \X : σ(y) = y }

(1.12) Proposition: (viz. 486)

(i) If X is a finite set, then the group (SX , ◦) is finite again, in fact itcontains precisely (#X)! elements, formally that is

#SX = (#X)!

40 1 Groups and Rings

(ii) Fix any 1 ≤ n ∈ N and denote the subgroup of all permutationsσ ∈ Sn+1 [of the set X := 1 . . . (n+ 1)] that fix n+ 1 by

S′n := {σ ∈ Sn+1 | σ(n+ 1) = n+ 1 }

For any k ∈ 1 . . . (n + 1) denote τk := (k n + 1) the transposition ofk and n + 1, i.e. the permutation that only interchanges these twonumbers. Then we will prove that the following subsets of Sn+1 areidentical

τkS′n :=

{τkπ | π ∈ S′n

}= {σ ∈ Sn+1 | σ(n+ 1) = k }

And thereby Sn+1 can be written as the following disjoint (i.e. for anyi 6= j ∈ 1 . . . (n+ 1) we have τiS′n ∩ τjS′n = ∅) union

Sn+1 :=

n+1⋃k=1

τkS′n

(iii) If (G, ◦) is any group, then G can be embedded into (SG, ◦). To beprecise the following map is well-defined and injective

L : G ↪→ SG : g 7→ Lg

Lg : G→ G : x 7→ gx

and a homomophism of groups, that is: Le = 11G and for any g, h ∈ Gwe get Lgh = LgLh. Nota in this sense the permutation groups SXare the most general groups whatsoever.

(1.13) Definition:Fix n ∈ N with n ≥ 1 and consider the perumation group (Sn, ◦). Then, forany permutation σ ∈ Sn and any i ∈ 1 . . . n, we define

fix(σ) := { i ∈ 1 . . . n | σ(i) = i }dom(σ) := { i ∈ 1 . . . n | σ(i) 6= i }

cyc(σ, i) :={σk(i) | k ∈ N

}`(σ, i) := #cyc(σ, i)

Thereby fix(σ) is called the fixed point set of σ and its complement dom(σ)is called the domain of σ. Also cyc(σ, i) is called the cycle of i under σand `(σ, i) is said to be the length of that cycle.

1.2 Permutations 41

(1.14) Proposition: (viz. 487)Fix n ∈ N with n ≥ 1 and consider the permutation group (Sn, ◦). Then,for any permutations % and σ ∈ Sn and any i ∈ 1 . . . n, we find the followingstatements

(i) As{σk | k ∈ Z

}is a subgroup of the finite group Sn we find that the

cycle cyc(σ, i) if finite, more precisely

`(σ, i) ≤ ord(σ) ≤ n!

(ii) The domain of σ is stable under σ, that is for any a ∈ dom(σ) we findthat σ(a) ∈ dom(σ) again, such that

σ(dom(σ)) = dom(σ)

In particular, if i ∈ dom(σ) is contained in the domain of σ, then σk(i)is contained in the domain of σ, for any power k ∈ N, such that

cyc(σ, i) ⊆ dom(σ)

(iii) If % and σ share a common domain D ⊆ 1 . . . n and agree on D, thenthey already are equal. Formally that is the equivalence

D := dom(%) = dom(σ)∀ a ∈ D : %(a) = σ(a)

}⇐⇒ % = σ

(iv) If the domains of % and σ are disjoint, then % and σ commute, formally

dom(%) ∩ dom(σ) = ∅ =⇒ %σ = σ%

(1.15) Definition:Fix n ∈ N with n ≥ 1 and consider an `-tuple (i1, . . . , i`) of natural numbersik ∈ 1 . . . n that are pariwise distinct (that is ij = ik =⇒ j = k). Then thetuple (i1, . . . , i`) gives rise to a permuation ζ ∈ Sn by letting

ζ(a) :=

a if a ∈ (1 . . . n) \ { i1, . . . , i` }ik+1 if a = ik for some k ∈ 1 . . . `− 1i1 if a = i`

This permutation ζ ∈ Sn is called a cycle of length ` and (by slight abuseof notation, as n is not specified) we will denote this cycle as

(i1 i2 . . . i`) := ζ

A cycle of length 2 is said to be a transposition of Sn. And by definitionit interchanges two numbers i 6= j. That is, if τ = (i j) then τ(i) = j andτ(j) = i and τ(a) = a for any other a ∈ 1 . . . n.

42 1 Groups and Rings

(1.16) Remark: (viz. 488)

• The notation of cycles is ambigious as it is of no effect with whichelement the cycle starts, as long as the ordering of elements is notchanged. That is (i1 i2 . . . i`) and (i2 i3 . . . i` i1) and so on until(i` i1 i2 . . . i`−1) all yield the same permutation. In particular wehave (i j) = (j i) for any transposition.

• Any cycle ζ can be written as a composition of transpositions. To beprecise: given the cycle (i1 i2 . . . i`) we can rewrite this one, as

(i1 i2 . . . i`) = (i1 i`)(i1 i`−1) . . . (i1 i2)

• Any transposition can be written as a composition of transpositionsof adjacent elements. To be precise given any i, j ∈ 1 . . . n with i < jwe can rewrite

(i j) = (i i+ 1)(i+ 1 i+ 2) . . . (j − 1 j)(j − 2 j − 1) . . . (i i+ 1)

• If ζ = (i1 i2 . . . i`) ∈ Sn is a cycle of length `, then for any otherpermutation σ ∈ Sn the conjugate σζσ−1 is another cycle of length `.In fact we get

σζσ−1 = (σ(i1) σ(i2) . . . σ(i`))

(1.17) Proposition: (viz. 489)Fix n ∈ N with n ≥ 1 and consider the perumation group (Sn, ◦). If nowσ ∈ Sn is any permutation with σ 6= 11 then the following five statementsare equivalent:

(a) σ is a cycle, that is there is some ` ≥ 2 and some pairwise disticti1, i2, . . . , i` ∈ 1 . . . n (that is ij = ik =⇒ j = k) such that

σ = (i1 i2 . . . i`)

(b) Any two elements a, b ∈ dom(σ) in the domain of σ are linked, i.e. thereis some k ∈ N such that b = σk(a).

(c) For any element i ∈ dom(σ) the domain of σ already is generated bythe orbit of i under σ, that is

dom(σ) ={σk(i) | k ∈ N

}(d) There is some i ∈ 1 . . . n such that the domain of σ is the orbit of i

under σ, that is

dom(σ) ={σk(i) | k ∈ N

}

1.2 Permutations 43

(e) If ` := ord(σ) denotes the order of σ, then there is some i ∈ 1 . . . nsuch that the domain of σ is given to be

dom(σ) ={σk(i) | k ∈ 0 . . . (`− 1)

}And in this case the length ` of the cycle σ = (i1 i2 . . . i`) is precisely theorder of σ, that is we get the following equality

ord(σ