Idempotent elements and ideals in group rings and the intersection

Sonderabdruck ana

ARCHlY DER MATtIEMATIK

Vol. XXIV. 1973 BLRKHAUSEll VERLAG, BASEL UND STUTTGART Faac. 6

Idempotent elements and ideals in group ringsand the intersection theorem

By

M. M. PARMENTERand S. K. SEHGAL

"

586 ARCH. MATH.

Idempotent elements and ideals in group ringsand the intersection theorem

By

M. M. PARMENTERand S. K. SEHGAL*)

1. Introduction. It is wen known [12] that the integral group ring ZG has no non-trivial (i.e. 9=0, 1) idempotents. It is also known [3] that KG the group ring of afinite group over an integral domain K has no nontrivial idempotents provided noprime p dividing I G I is invertible in K. Further Bovdi and Mihovski [2] have madethe following

Conjecture. Let K be a commutative ring with 1 having no nontrivial idempotents.Suppose that no element of the group G has order invertible in K. Then KG hasno nontrivial idempotents.

Actually, they conjectured this for int~gral domains K and supported it by aresult for nilpotent groups. We offer in support of this conjecture proofs in case Gis nilpotent or supersolvable (if 2 is not invertible in K) or finite solvable.

. Related to this is the question of existence of idempotent ideals in ZG. Akasaki [1]has proved that if G is finite nilpotent then ZG has no nontrivial (i.e. 9=0, ZG)idempotent (two sided) ideals. He has pointed out that if G is finite and not solvablethen ZG has a nontrivial idempotent ideaL Further, he has conjectured that for afinite group G, ZG has no nontrivial idempotent ideals if and only if G is solvable.'Ve prove (Theorem 4.4) that LJ(G) the augmentation ideal of ZG, G finite group,contains no nonzero idempotent ideals (of ZG) if and only if G is solvable. Of course,the possibility remains that there exist idempotent ideals not contained in L1 (G).This result is a consequence of Theorem 3.10 which essentially says that a finitegroup G is solvable if and only if a certain sequence of ideals associated with LJ(G)(as defined in section 3) eventually vanishes.

One can not expect Akasaki's result to hold for arbitrary (nilpotent) groups be-cause it is well known [5] that if G is an abelian torsion divisible group then

LJ2(G) = LJ4(G) 9=O.

We extend Akasakj's result to finitely generated nilpotent groups. This is a con-sequence of Lemma 4.2 which is classical for commutative Noetherian rings.

*) This author is recipient of an Alexander von Humboldt Fellowship at the MathematischesInstitut of the university of Heidelberg. This work is supported in part by N.R.C. Grant No.A-5300.

Vol. XXIV, 1973 Idempotent elements and ideals in group rings 587

Another consequence of Lemma 4.2 is that if ZG is the group ring of a finitelygenerated nilpotent group then there exists an i E A (G) with

.This is also contained in the work of Smith [15]. We investigate this further andgive (Theorem 5.5) necessary and sufficient conditions for the result to hold if G isfinitely generated with a torsion element. We also give (Theorem 5.6) necessary andsufficient conditions for such an i to exist if G is finitely generated and the coefficientring is tp, the ring of p-adic integers.

We shall denote by AK(G) the augmentation ideal of KG and shall often dropthe subscript K. The kernel of the natural homomorphism KG ~ K(GjN), for anormal subgroup. N will be denoted by AK(G, N). It is equal to AK(N)KG. Weshall assume that the ring K has an identity.

2. Idempotents in KG. In order to prove our first Lemma we need the followingwell known intersection theorem of Krull.

(0An (G») (1 - i) = O.

Theorem 2.1. Let ] be an ideal of a commutative Noetherian ring R. Then thereexists an i E] such that

](1)(1 - i) = 0, where ](1)= n]n.n

Lemma 2.2. Let K be a commutative ring without nontrivial idempotents such that anatural prime

-p times-

p = (1 + 1 + ... + 1)

is not a unit in K. Suppose that N is a finite normal p-subgroup of G such that K(GjN)has no nontrivial idempotents. Then KG has no nontrivial idempotents.

Pro 0 f. Let e = e2 = Leg g E KG. By considering 1 - e if necessary we may as-sume that e E A (G).We may also assume that K is a Noetherian ring. Since K(GjN)has no idempotents e is an element of A (G, N). Therefore,

e En An(G, N) = A(I)(G,N) = A(I)(N)KG.n

The last equality is a consequence of the fact that

KG= Lx"KN,"

where {x,,} is a transversal of N in G.Since p is not invertible in K, K = KjpK is a ring of characteristic p. We con-

clude that e = 0 in KG. Let ] be the ideal generated by {eg}. Then] = ]2 and

] < pK. By Theorem 2.1 there exists i E ] such that

](1)(1 - i) = 0 Le. ](1 - i) = O.

Hence we have i2 = i and therefore i = 0 since i = 1 will imply that] = K. Wehave thus] = 0 and hence e = O. .

588 M. M. PARMENTER and S. K. SEHGAL ARCH. MATH.

Theorem 2.3. Suppose that G is a supersolvable group and that K is a commutativering without nontrivial idempotents in which the order of no group elements =f:1 is in-vertible. Suppose further that 2 is not invertible in K. Then KG has no nontrivialidempotents.

Proof. Since G is supersolvabfe, it has a characteristic series 1 < L < M < Gsuch that L is finite of odd order, MIL is finitely generated torsion free nilpotentand GIM is a finite 2-group [13]. We prove first that KG has no nontrivial idem-potents if L = 1. Suppose that we have

e = e2E L1 (G) and 1 < M < G

where GIM is a finite 2-group and M is torsion free nilpotent. By Lemma 2.2

e En L1n (G, M) = L1w(M)KG.n

But

L1w(M) = 0

by [4] and hence e = O.We may, therefore, conclude that K (GIL)has no nontrivialidempotents.

Suppose that N 9=1 is a minimal normal (in G) subgroup of L such that K (GIN)has no nontrivial idempotents. Since N' 9=N'we can find a characteristic subgroupT of N such that NIT is a p-group. We conclude by Lemma 2.2 that K (GIT) hasno nontrivial idempotents. This contradicts the minimality of N. Hence N is trivialand K (G) has no nontrivial idempotents.

The argument given above also proves

Theorem 2.4. Suppose that G is a (locally), nilpotent or finite solvable group andthat K is a commutative ring without nontrivial idempotents in which the order of nogroup element 9=1 is invertible. Then KG has no nontrivial idempotents.

For our next result we need to investigate conditions for the vanishing of trans-finite powers of the augmentation ideal.

3. Transfinite "powers" of L1(G) for solvable groups. We shaH define certain trans-finite powers of L1 (G). These are not powers in the usual sense of the word but ratherlike operators and no confusion is likely to arise in our context. We shall write forL1 = L1 (G),

L1w= n L1n and by induction L1wm= (L1wm-l)Wn

for natural numbers m.Write

L1ww= n L1wm.m

We shall write

itA for L1ww.

Further define by induction for natural numbers m

L1Am= (itAm-l)A.

VoL XXIV, 1973 Idempotent elements and ideals in group rings 589

The following is Theorem E of Hartley [6] and will be used repeatedly by us.

Theorem 3.1. Let G be a group having a finite strong central series, each factor afwhich is a direct product af a free abelian graup and a graup of exponent dividing pn,where p is a fixed prime and n a fixed integer. Let K be a commutative ring satisfying

npnK = o.n

Then

LlW(G)= o.

IJemma 3.2. Suppose that G has a sequence af normal subgraups {Ni}

N1 JN2J... JNiJ ... with nNi = 1.i

Then

nLt(G,Nt) =0.i

Proof. Obvious.

An immediate consequence of the last two results is the

Lemma 3.3. Let P be an abelian p-graup with nO'elements ( =1= 1) af infinite p-height(i.e. n ppn = 0). Let K be a cammutative ring satisfying

n

npnK=O.n

Then

LtW(P) = o.

Lemma 3.4. Suppase that G is an abelian tarsian graup and that K is a commutativering. Suppase that far any prime p far which the Sylaw p-group Sp(G) =1= 1, we have

npnK=On

and that Sp (G) has no element ( =1= 1) af infinite p-h.eight. Then

LtA(G) = o.Proof. Write

G = G1 J G2 J G3 J . . .

such that GdGi+1 is a pi-group and Pi =1= Pi for i =1= j. By the last Lemma and induction

Ltwn (G) c Lt (G, Gn+1) .Since

nOn = 1,n

by Lemma 3.2 we can conclude that.

LtA(G) = O.


The next is a theorem of Jennings [7] as proved by Formanek [4]. .

Lemma 3.5. Let G be a finitely generated torsion free nilpotent group. Then for. anycoefficient ring K, the augmentation ideal .1(G) of KG satisfies

.1(0(G) =0.

It is possible to remove the condition of finite generation of G in the above if Kis the ring of rational integers Z [6].

For the abeIian case, one can prove the

Lemma 3.6. Let G be a torsion free abelian group and let K be an integral domainof characteristic O. Then

.1(0(G) =0.

For the proof, it suffices to establish:

If A is a subgroup of a finitely generated torsion free abelian group Band K isan integral domain of characteristic 0, then

(*) LI~(A) = LI~(B) n LIK(A).

We omit the proof of (*) which can be obtained by direct computation.

Remark. It has been pointed out by Formanek [4] that the above Lemma doesnot hold for characteristic p > o.

Lemma 3.7. Suppose that G is a finitely generated abelian group. Suppose that K isa commutative ring satisfying

npnK=On

for every prime for which Sp (G)4=1. Then'

LIwk(G) = 0,

for some integer k.

Proof. Lemma 3.5 and Lemma 3.4.

We are now ready to extend these considerations to solvable groups.

Theorem 3.8. Let G be a solvable group such that each factor in the derived series isfinitely generated. Let K be a commutative ring satisfying

npnK=On

for each prime p occuring in the factor groups of the derived series of G. Then in KGwe have

LIAC(G) = 0,

where c is the length of the derived series of G.


Proof. Let

0 = OOJGI J "."JOe-IJ Ge = 1

be the derived series of G. If c = 1, the theorem follows from Lemma 3.7. Usinginduction we have that

LPC-l(OIGe-I) = 0I.e.

jAC-l(G)cj(G,OC-I) = j(Ge-I)KG.Now,

jAC(G)c (j (Ge-I) KG»).= j).(OC-I) KG = 0

and the result is proved.

Theorem 3.9. Let 0 be a solvable group such that the Sylow p-subgroups of the factorgroups in the derived series have no nontrivial elements of infinite p-height. Let K bean integral domain of characteristic 0 satisfying

() pn K = 0n

for each prime p occuring in the factor groups of the derived series. Then in KG we have

. jAC(G) =0,

where c is the length of the derived series of o.

Proof. Induction on c and for c = 1 Lemma 3.3 and Lemma 3.6.

Remar k. It isknown ([5], Page 58) that ifG is a group whose factor commutator

group is torsion divisible and K is the ring Z of integers then j2 = j3. Hence a

condition on the torsion subgroups of the factor groups in the derived series as in

the last two theorems is required.

Theorem 3.10. Let G be a finite group and let K be a commutative ring satisfying

()pnK = 0n

for all primes p dividing the group order I GI. Then

jAC(G) = 0

for some c if and only if 0 is solvable.

Proof. If 0 is solvable then

jAC(G) = 0

by Theorem 3.8. If 0 is not solvable it contains a nontrivial normal subgroup Nwhich is its own derived group. Then

A2(N) = A (N) c j (G)and hence

0 '*'L1(N) CAAC(0) fol' all c.


4. Idempotent ideals in KG. It has been proved by Roseblade [14] that any idf'-alin the integral group ring of a finitely generated nilpotent group has a centralizingset of generators in the following sense.

Definition 4.1. A set of generators {Xl, ..., Xu} of an ideal of a ring R is said tobe a centralizing set of generators if

(i)

(ii)

Xl E G(R), the center of Rand

Xi + (Xl, ..., Xi-I) EO (Rj(xI, ..., Xi-I)) for 2 < i < n.

Here (Xl, ..., Xi-I) denotes the ideal of R generated by Xl, ..., Xi-I.A classical argument "gives the following Lemma which we shall also need in the

next section.

Lemma 4.2. Let ZG be the integral group ring of a finitely generated nilpotent group.Let A and M be ideals of ZG satisfying A c AM. Then there exists an element m of Msuch that (1 - m)A = O. . .

Proof. Let {Xl, ..., xu} be a centralizing set of generators of A written in thereverse order. Write

Ai = (Xi, ...,xu) for 1<i<nand

AU+1 = O.

We shall produce by induction elements mi EM such that

(1 - mi)A cAi.

Then mu+1 will be the required element m. For i = 1, take mi = O.Suppose we already have found mI, ..., mi with the required property. Then

AcAM and (1-mdAcAi gives that

(1 - mi)A c (1 - mi)AM c AiM c MAi mod Ai+1.Thus, u

(1 - mi)xi = LmijXj for some milE M.j=i

Hence

(1 - mi - mti) Xi EAi+l

and we may take

1 - mi+l = (1 - mi - mti) (1 - mi).

This proves the Lemma.The next Theorem has been proved by Akasaki [1] for finite nilpotent groups.

Theorem 4.3. Let G be a finitely generated nilpotent group. Then ZG contains no non-trivial idempot~nt ideals.

Proof. Let I be an idempotent ideal in ZG. By taking

A=M=I


in the last Lemma, there exists an i E [ such that (1 - i) [ = O.Hence i2 = i and(by Theorem 2.4 for example) i = 0 or 1. Hence [= 0 or ZG.

Remark. Some condition on the nilpotent group is necessary for the absence ofidempotent ideals in ZG. This can be seen from the fact that if G is a divisible torsionabelian group then

Ll2= Ll3= Ll4 [5] .

One can also prove the following theorem by induction using an argument similarto Lemma 2.2.

Theorem 4.4. Let G be a finite group. Let K be a commutative ring satisfying

(lpnK=On

for all primes p dividing I G I. Then LlK(G) contains a nonzero idempotent ideal if andonly if G is not solvable.

Proof. Let G be finite solvable and let

[2 = [ c LI (G) .Then

[ C LlAC(G) for all cand hence

[=0

by Theorem 3.10. Conversely, as pointed out by Akasaki [1], if G is uJt solvablethere exists a nontrivial perfect normal subgroup N of G. Then 0 =1=L1 (N) = Ll2(N)and therefore

LI(G, N) = Ll2(G,N).

Remark. Let G and K be as in Theorem 3.8 or Theorem 3.9 then LlK(G) containsno nontrivial idempotent ideals (of KG).

5. The intersection theorem. The next theorem is a consequence of Lemma 4.2and the Artin-Rees property which is known to hold (see [9], [14]) in ZG when Gis finitely generated nilpotent.

Theorem 5.1 ([15], [14], [9]). Let LI(G) be the augmentation ideal' of ZG where G isa finitely generated nilpotent group. Then there exists an i ELI (G) such that

LlW(G)(1 - i) = O.

In this section we shall prove this theorem for more general coefficient rings andshall also find necessary and sufficient conditions for such a theorem to hold forcertain, groups G.

Lemma 5.2. Let K be a commutative ring and let

G1 X G2 X ... X Gn

ArdJiv der Mathematik XXIV 38

594 M. M. PARMENTER and S. K. SEHGAL ARCH. MA THo

be a nor'mal sub(/roup of B with each Gt normal in B 0 Then

iJ (B, G1 X ... X Gn-1) n iJ (B, G1 X ... X Gn-2 X Gn) n ...... n iJ (B, G2 X ... X Gn) = 2:LJ(B, Gi)LJ(B, Gj).

. i*i

Proof. The proof proceeds by induction on n. First we take n = 2. Certainly

iJ (B, G1) iJ (B, G2) c iJ (B, G1) n iJ (B, G2) .

Conversely, let x E iJ (B, G1) n LJ(B, G2). We can write

x = 2:(J.ijhl, (/1£(/2i «(/l,j - 1)i, i

where (XtjE K, (/1£E G1, (/21,E G2, (/lj E G1 and the ht belong to a fixed transversalof G1 X G2 in B. Now,

x = 2:(Xii hi (/It «(/u - 1) «(/lj - 1) + 2: (Xij hi (/1£ «(/lj - 1) 0

i, i i, i

Since x E iJ (B, G2) the second sum in the expression for x belongs to iJ (B, Gz) andtherefore is zero. Hence,

X E iJ (B, G2)LJ(B, G1)= iJ (B, G1)iJ (B, G2) .

Now assume that the Lemma holds for n -1. Hence,

By going from B to B = BIG1, we conclude that

iJ (H, G1 X ... X Gn-1) n iJ (B, G1 X ... X Gn-2 X Gn) n ...... n LJ(B, G2 X ... X GnH~

~ iJ (H, G1) + 2:iJ (B, Gi) iJ (H, Gj).i*i

i,i~2

iJ (B, G2 X ... X Gn-l) n iJ (H, G2 X ... X Gn-2 X Gn) n'"

... n LI(H, G3 X ... X Gn) = 2:LJ(B, Gi) LI(B, Gj).i*i

i,j~2

We observe that

2:LI (B, G i) iJ (B, Gj) ~ iJ (H, G2 X ... X G n)i*i

i,i~2

and by induction.

iJ (B, G1)n LI(B, G2 X ... X Gn) = iJ (H, G1) iJ (B, G2 X ... X Gn).

Therefore

LJ(B, G1 X ... X Gn-1) niJ (H, G1 X ... X Gn-2 X Gn) n'"

... n iJ (B, G2 X ... X Gn) ~

~(LJ(H, G1) + ?LI (H, Gi)iJ (H, Gj)

)n LI(H, G2X ... X Gn) =

1*~i,i~2 .

= iJ (B,G1) iJ (G2X ... X Gn) + 2:iJ (H,Gi) LJ(H,Gj) ~ 2:iJ (B,Gt) iJ (H,Gj).i*i j*i

i.j~2 i,i~2


The last inclusion follows because

A (H,G2 X ...X Gn) ~ 2:LI (H, Gi)i~2

due to the identity

gh - 1 = (g- 1) + (h - 1) + (g - 1)(h - 1).

The containment in the other direction being trivial, the Lemma is proved.

Theorem 0.3. Let G be a finitely generated nilpotent group and let K be a cmnmutativering with the property that if there exists an element of prime order p in G then either pis a unit of K or

npnK = o.n

Then there exists an element i of Lt(G), the augmentation ideal of KG, such that

LtW(G)(1 - i) = O.

Proof. We use induction on the order of T (G) the torsion subgroup of G. IfT (G) = 1 then

LlW(G)= 0 [4] and i = 0will do.

If T (G) = P is a p-group then by a theorem of Gruenberg ([8], Page 80) G isresidually a finite p-group and hence [6]

LtW(G)= 0

providednpnK = O.

n -

If p is invertible then since

LlW(G)c Lt(G, P)

we may tak~ - -I PI~l (2: X).

~ - 1 zeP

Now assume that there exist at least two different primes-p and q, dividing I T (G)I.Choose Zl, Z2 central in G such that

O(Zl) = p and O(Z2) = q.

We first make the following observation. There exist integers a and b such thatap + bq = 1 and therefore,

(Zl - 1) (Z2 - 1) = (ap + bq) (Zl - 1) (Z2 - 1) == ap(zl - 1) (Z2 - 1) + bq(Zl - 1) (Z2 - 1) =

= - a (~) (Zl-1)2(Z2 -1) - ... - a(zl-1)P(Z2 -1)-

- b (~) (Zl-1) (Z2-1)2 - ... - b(Zl-1) (Z2- l)Q.'112*


Hence,

0 = (Zl - 1) (Z2- 1) {1 + a (~) (Zl - 1) + ... + a(zl - 1)p-l +

+ b (~) (Z2 - 1) + ... + b (Z2 - 1)q-l}.

Thus there exists an element j E Lf (G) such that

(Zl - 1) (Z2 - 1) (1 - j) = O.

By induction on I T (G)I, there exist il, i2 E Lf (G) such that

LfW(G/<Zl» (1 - it) = 0 and Lfw(G/<Z2»(1 - i2) = O.Hence,

LfW(G)(1 - it) (1 - i2) k Lfw(G)n Lf(G, <Zl» n Lf(G,<Z2» kk LfW(G) n (Lf (G, <zl»Lf (G, <Z2» ,

by Lemma 5.2.Also by the observation that we made above, there exists j ELf«Zl> X <Z2» such

that

(Zl - 1) (Z2- 1) (1 - j) = O.

Therefore,LfW(G)(1 - it) (1 - i2) (1 - j) = 0

and the result holds with 1 - i = (1 - it) (1 - i2)(1 - i).

Corollary 5.4 [15]. Let KG be the group ring of a finitely generated nilpotent group Govera commutativeNoetherian integral domain K. Let Lf(G) be the augmentation idealof KG. Then there exists an i ELf (G) such that

Lfw(G)(1 - i) = O.

The next theorem gives a necessary and sufficient condition for the intersectiontheorem to hold for Lf(G) in the integral group ring of a finitely generated group witha torsion element.

Theorem 5.5. Let G be a finitely generated group with a nontrivial torsion element andlet fJ = {pip a prime and there exist8 an element g EG of order p}.

Then the following are equivalent:

(i) There exists i ELfz(G) such that Lfz(G) (1 - i) = O.

(ii) For all p E fJ, nDn,ip (G) is finite of order relatively prime to p.n

Here, Zp denotes the ring of p-adic integers. And for any ring K, Dn.K(G) standsfor the n-th dimension subgroup over K namely,

Dn.K(G) = G n 1 + Lf~(G).

Proof. First assume that (i) holds. If

tEn Dn,z(G)n


then

and so(t - 1)(1 - i) = 0

n Dn.z(G)n

is finite and .( "",g )Z G .

1-'tE L.. G)g E ('\ Dn.z(n

This implies that the content (sum of coefficients) of 1- i is divisible by

I nDn.z(G)1n

in Z. So

nDn.z(G) = 1.n

Let g E G be of order p and let

hEn Dn,zp (G) .n

Then, because the order of h modulo Dn.z(G) is relatively prime to p (Theorem 5.1[11]),

(h - 1) (g - 1) E n LI~(G)n

and hence

(h - l)(g - 1)(1 - i) = O.

Since the content of 1 - i is 1, (g - 1) (1 - i) *0 and hence

nDn,zp (G)n

is finite. Since

nDn,z(G) = 1,n

it follows (by Theorem 5.1 [11]) that

nDn,zp (G)n

is p-torsion free.Now assume that (ii) holds. Since GIDn.zp (G) is finitely generated nilpotent and

contains no elements of order relatively prime to p it follows by a theorem of Gruen-berg ([8], page 80) that GIDn.zp(G) is residually a finite p-group. Hence

GIn Dn,zp (G)n

is residually a finite p-group. Therefore by Hartley's Theorem E [6]

(**) nLl~(G) ~ nLlz (G, nDn,zp(G» ).

m p n


Now nDn.zp(G) is finite and v-torsion free by assumption (for p E (J). Ifn

of prime order q *p. The torsion elements of G form a finite normal subgroup Twhere. .

T = n Dn,zp (G) .n Dn. z"(G) .n n

. T is also nilpotent since

nnDn.zp(G) = 1pep n

and thus

nDn,z(G) = 1.n

Let T = SI X ... X Se, where St is the pt-Sylow subgroup of T. Noting that

nDn.ip (G)= TISf,n 10 i*k

we get by Lemma 5.2

nLlz(G, nDn.zp(G)) = 2:Llz(G, Sk) LIz(G,Sj).p n i*k

Hence, from (**) we have

n LI~(G) ~ 2:Llz(G, Sk)Llz(G, Sj).m i*k

By Corollary 5.4, there exists an i E Llz(T) such that (1- i) annihilates

Llz(T) = 2: Llz(T, Sk)Llz(T, Sj).i*k

Since Sj are normal in G for all j, we have LIz (G)(1- i) = O.This completes the proofof the theorem.

In the next theorem we give a necessary and sufficient condition for the inter-section theorem to hold if G is finitely generated and K = tp.

Theorem 5.6. Let G be a finitely generated group. Then the following are equivalent.

(i) There exists an i E Llzp(G) such that

Llzp(G) (1 - i) = O.

(ii) nDn.zp (G) is finite and of order rekltively pri'lM to p..n

n Dn.zp (G) = 1nthen

n LI(G) = O.n

If

nDn.zp (G) * 1 for some p E (Jn

then choose

hEnDn.zp(G)n

Vol. XXIV. 1973 Idempotent elements and ideals in group rings 599

Proof. For simplicity we omit tp from the subscripts of LI and D in the courseof this proof. Assume first that (i) holds. Then (g-1)(1-i) = 0 for each

gEnDn(G)n

and so n Dn(G) is finiteandn

(1-i)E ( L g)tpG.g € ("\ D,,(G)

"

Hence the content 1 of 1 - i is divisible by I nDn (0) I and we can conclude thatI nDn (G) Iis relatively prime to p. n

n

Now assume that (ii) holds. It is easy to check that

() Dn(G/{)Dn(G» = 1and hence by a result of Parmenter ([10], Theorem 7)

() Lln(G/{)Dn(G») = O.Hence

Since0L1n(G) ~LI(0. f)Dn(G»).

nDn (G)n

is finite and of order relatively prime to p, we have for any g EnDn (G)n

(In Dn (G)

\-1 L g)(g - 1) = 0

n g€("\Dn(G)n

and the theorem holds with

i=1-lnDn(G)I-l L g.n g€("\Dn(G)

n

The next result gives a necessary and sufficient condition for the intersection theoremto hold in group rings of finite groups over integral domains.

Proposition 5.7. Let G be finite and K a rommutative integral domain. Then the follow-ing are equivalent.

(i) There exists an i ELlK(G) such that Llx(G)(1 - i) = O.

(ii) InDn,K(G)1 is invertible in K and any prime p dividing IGI is invertible orn

satisfies n pn K = O.n

Proof. Assume that (i) holds. Since (g - 1)(1 - i) = 0 for any g E n Dn.K(G)n

it follows that In Dn,K(G) I is invertible. Let-g E G be of order p and let 0 =F r En

E npnK. Because r(g - 1) E nLln(G) we have r(g - 1)(1 - i) = O. Thereforen n

(g - 1) (1 - i) = 0 and hence p is invertible.


Now assume that (ii) holds. Since G is finite,

nDn,K(G) = Dm,K(G)n

for some m. Let

H = GjDm,K (G).

Then H is a finite nilpotent group. Therefore by Theorem 5.3 we have l E LlK(H)such that

LlK(H)(1 -f) = o.

Consequently,

LlK(G) (1 - t) c LlK(G,Dm,K(G)), t E LlK(G).

Since IDm,K(G) I is invertible in K we see as in the last Theorem that there existsa j E LlK(G) such that

LlK(G)(1 - t)(1 - j) = O.

This completes the proof with 1 - i = (1 - t) (1 - i).

Re(erencf.S

[1] T. AKASAKI, Idempotent ideals in integral group rings. J. Algebra 23, 343-346 (1972).[2] A. A. BOVDI and S. V. MmovsKI, Idempotents in crossed products. Dokl. Akad. Nauk

SSSR 195, Soviet Math. Dokl. 11, 1439-1442 (1970).[3] D. B. COLEMAN,Idempotents in group rings. Proc. Amer. Math. Soc. 17,962 (1966).[4] E. FORMANEK,A short proof of a theorem of Jennings. Proc. Amer. Math. Soc. 26,405-407

(1970).[5] K. W. GRUENBERG,Cohomological topics in group theory. Lect. Notes in Math. 143, Berlin

1970.[6] B. HARTLEY,The residual nilpotence of wreath products. Proc. London Math. Soc. (3) 20,

365-392 (1970). .[7] S. A. JENNINGS, The group ring of a class of infinite nilpotent groups. Canad. J. Math. 7,

169-187 (1955).[8] H. NEUMANN,Varieties of Groups. Berlin 1968.[9] Y. NouAZE and P. GABRIEL, Ideaux premiers de l'algebre enveloppante d'une algebre de

Lie nilpotehte. J. Algebra 6, 77-99 (1967).[10] M. M. PARMENTER,On a theorem of Bovdi. Canad. J. Math. 23, 929-932 (1971).[11] M. M. PARMENTER,I. B. S. PASSI and S. K. SEHGAL,Polynomial ideals in group rings.

Canad. J. Math. (to appear).[12] D. S. PASSMAN,Infinite group rings. New York 1971. .

[13] D. J. S. ROBINSON,Finiteness conditions and generalized soluble groups. Berlin 1972.[14] J. E. ROSEBLADE,The integral group rings of hypercentral groups. Bull. London Math.

Soc. 3, 351-355 (1971).[15] P. F.SMITH, On the intersection theorem. Proc. London Math. Soc. (3) 21, 385-398 (1970).

Eingegangen am 20.3.1973

Anschrift der Autoren:

M. M. ParmenterDepartment of MathematicsMemorial University of NewfoundlandSt. John's, Canada

S. K. SehgalDepartment of MathematicsUniversity of AlbertaEdmonton, Canada

Idempotent elements and ideals in group rings and the intersection

Documents

Transcript of Idempotent elements and ideals in group rings and the intersection