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
76 Bertalan Király We axe interested in the finiteness of the terminais of groups. The groups with finite terminais with respect to integers are well known and easily described (see [1]). In this case the terminais of groups are 1 or 2. We are primarily concerned with fin ding all groups whose terminais with respect to commutative rings with unity are finite and with describing the terminais of such groups. In this paper we give necessary and sufficient conditions for groups which have fmite terminais with respect to a commutative ring with unity of non-zero characteristic (Theorem 3.3). In Theorem 3.4 we give the qualitative characterisation of T r(G) using the ring-theoretical terminology. 2. Notations and somé known facts If H is a normal subgroup of G, then I(RH) (or I(H) for short when it is obvious from the context what ring R we are working with) dénotés the ideal of RG generated by all elements of the form h — l,(h G H). It is well known that I(RH) is the kernel of the natural epimorphism 4> : RG —> RG/ H induced by the group homomorphism (f) of G onto G/ H. We notice that if H = G then I{RG) = A(RG). If /C dénotés a class of groups (by which we understand that /C contains all groups of order 1 and, with each H G /C, all isomorphic copies of H), we define the class R/C of residually-/C groups by letting G G R/C if and only if : whenewer 1 / g G G, there exists a normal subgroup H g of the group G such that G/H g 6 K, and g H g We use the following notations for standard group classes: Af 0 - torsionfree nilpotent groups, Af p - nilpotent p-group s of finite exponent, that is, nilpotent group in which every element g satisfies the équation g p = 1 for some n = n(G). Let /C be a class of groups. A group G is said to be discriminated by K, if for every finite subset g\ , . .., g n of distinct elements of G, there exists a group H G /C and a homomorphism 0 of G into H such that 4>{g l) ^ 4>(gj) for ail ^ / gj, (1 <i,j< n). Lemma 2.1. Let a class K, of groups be closed under the taking of subgroups (that is ail subgroups of any member of the class /C are again in the class KL) and also finite direct products and let G be a residually—IC group. Then G is discriminated by JC. The proof can be obtained easily. It is easy to show that if G is discriminated by a class of groups JC and if x is a non-zero element of RG , then there exists a group H G /C and a homomorphism (f) of RG into RH such that <fi{x) ^ 0.
