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 79 Since I(-y k(G)) Ç A k(RG)we have y k - 1 G A k(RG). Therefore h- 1 = (/if /if 1) (mod rt n r> n nApplying (2) repeatedly to (h\ h 2 • • • h^ -1) from the preceding expression it follows that m m p n , ns h- 1 = ]T(/*f - 1)6, = £ Y, [• ) - lY bi ( mo d where G Í2G. The elements — l)7 are in A k(RG ) for ail i and j > k. fp n\ If n > s -f k, then p s divides I 1 for ] — 1,2,..., k — 1. Therefore 771 fc —1 (3) /I - 1 = p s Y, dÁ hi - - P^CO (mod A*(ÄG)), 2 = 1 J = 1 m k — l where F k(h ) = ^ ^^ dj(hi — l) Jbi and p sdj = 2 = 1 j = i we have that /i-l G A k(RG) which implies the inclusion I{G Pj k) Ç A k(RG)) and complétés the proof of the lemma. Lemma 3.2. Let R be a commutative ring of characteristic p s . Then A U(RG) = I(G P). PROOF. From Lemma 3.1 the inclusion I(G P) Ç A U(RG) follows. We can readily verify that G/G P is residually-A/^ group and so .A U(RG/G P) = 0 by Theorem 2.3. Hence we have the inclusion A L J{RG) Ç I(G P ) which complétés the proof of the lemma. If G is a finite p-group and R a commutative ring of characteristic p s , then the ideal A(RG) is nilpotent (see Theorem 2.1). Dénoté by R°(A(RG)) the nilpotency index of A(RG ), i.e. the natural number k = T°(A(RG)) for which A K~ L(RG) ï A K(RG) = 0. If G = (1> we put R°(A(RG)) = 1 Let r p(G) dénoté the smallest natural number k (ifit exists) such that T G P TK — ... — G P. . Since p s is zéro in R
