Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

KIRÁLY, B., On the powers of the augmentation ideal of a group ring

Ori the powers of the augmentation ideal of a group ring 77 From this fact we have Lemma 2.2. If G is discriminated by a class of groups /Cand for each H E IC the équation A"(RH) = 0 holds, then A"(RG) = 0. If K

, L are two subgroups of G, then we shall denote by [K, L] the subgroup generated by all commutators [g,h] = g"

1 h~

lgh, g £ K

, h E L. A sériés G = Ci D G

2 2 • • • 2 G

n D ... of normal subgroups of a group G is called an TV-series if [Gi,Gj] Ç Gi+j for all i,j > 1 and also each of the Abehan groups Gi/Gj is a direct product of (possibly infinitely many) cyclic groups which are either infinite or of order p

k , where p is a fixed prime and k is bounded by some integer depending only on G. It is easy to see that the lower central sériés of a nilpotent p-group of finit e exponent is an N-sériés. In this paper we shall use the following theorems: Theorem 2.1. ([5] Lemma 2.21, page 27) The augmentation ideal A(RG) is nilpotent if and only if G is a finite p-group and R has characteristic p

n for some prime p. The ideal J

V(R) of a ring R is define d by oo jp(R)

=n p

n

R

• 71=1 Theorem 2.2([3]). Let G be a group having a finite N-series and R be a commutative ring with unity satisfying J

P(R

) = 0. Then A

U(RG) = 0. In this paper we apply Theorem 2.2 for residually-A/^ groups. Theorem 2.3. Let R be a commutative ring with unity satisfying J

P(R) = 0. H G is a residually-Ai

p group, then A

U(RG) = 0. The proof of this theorem follows from Lemmas 2.1 and 2.2 and Theorem 2.2 because the class Ai

p is closed under the taking of subgroups and also finite direct products. Theorem 2.4 ([4],VI.,Theorem 2.15). If G is a residually torsion-free nilpotent group and R is a commutative ring with unity such that its additive group is torsion-free, then A

U(RG) = 0.