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
80 Bertalan Király Theorem 3.1. Let R be a commutative ring of characteristic p s Then the augmentation terminal of G with respect to R is imite if and only if G jG v is a finit e p-group. PROOF. By Lemma 3.2, T r(G) = 1 if and only if G = G P. Now we suppose that T r{G) = k > 1. Then ... D A k~ 1(RG) D A k(RG) = A k+ 1(RG) = ... = A u ;(RG) and hence ... D A k~ 1(RG/G p,i) D A k{RG/G p> i) = ... = A u(RG/G Pf i), that is T R(G/G Pl i) is finite and not greater than T r(G) for ail i > 1. It is very easy to see that G/G P ti are residually-A/^ groups and consequently, by Theorem 2.3, A U{RGJG P, %j = 0 for ail i > 1. Because R R(G/G P) I) < K , (4) A k(RG/G Pj i) = 0 for every i. So from the isomorphism A k(RG/G P ti) * (A k{RG ) + I{G Pf i))/I(G Pi i) the inclusion A k{RG) Ç I{G p,i) follows for ail i > 1. If i = k then from Lemma 3.1 it follows that A k(RG ) = I(G PHence I(G p,k) Q I{G Pand, therefore,G P ifc Ç G p_i for ail i > 1. This implies that ... D G P ik = G P ik+ 1 = ... = G p and from (4) we have that A k(RG/G p) = 0. So, by Theorem 2.1, G/G p is a finite p-group. Conversely, let G/G p be a finite p-group. Then, by Theorem 2.1, A k(RG/G p) = 0 for the nilpotency index r°(A(RG/G p)) = k. It follows that A k(RG) Ç I(Gp). Hence, by Lemma 3.2, we obtain that A k(RG) Ç A U'{RG). The inverse inclusion, of course, is trivial. Therefore A k(RG) = A U(RG). Consequently A k{RG) = A k+ l(RG) which was to be proved. Theorem 3.2. Let R be a commutative ring of characteristic p s and let the augmentation terminal of G with respect to R be finite. Then T r(G) = r°(A(RG/G P)) > r P(G).
