Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)
KlRALY, B., Residual Lie nilpotence of the augmentation ideal
Residual Lie nilpotence of the augmentation ideal BERTALAN KIRÁLY* Abstract. In this paper we give necessary and sufficient conditions for the residual Lie nilpotence of the augmentation ideal for an arbitrary group ring RG except for the case when the derived group of G is with no generalized torsion elements with respect to the lower central series of G and the torsion subgroup of the additive group of R contains a non-trivial element of infinite height. From this results we get the residual Lie nilpotence of the augmentation ideal of the p-adic integer group rings. 1. Introduction Let J? be a commutative ring with identity, G a group and RG its group ring. The group ring RG may be considered as a Lie algebra, with the usual bracket operation. The study of this Lie algebra Was initiated by I. B. S. Passi, D. S. Passman and S. K. Sehgal [5]. Additional results on the Lie structure of RG may be found in [4] and [6]. Let A(RG) denote the augmentation ideal of RG, that is the kernel of the homomorphism RG onto R which sends each group element to 1. It is easy to see that as ß-module A(RG) is a free module with elements g — 1 (g £ G) as a basis. There are many problems and results relating to A(RG) ([4], [6]). In particular, it is an interesting problem to characterize the group rings whose augmentation ideal satisfy some conditions. In this paper, we treat the Lie property. The Lie powers A^(RG) of A(RG) are defined inductively: A^(RG) = A(RG), A^+ l^(RG) = [AW(RG),A(RG)}RG,i f Ais not a limit ordinal, and for the limit ordinal A, A^(RG) = f] u< xA^(RG), where [K,M] denotes the i?,-submodule of RG generated by [k,m] = km — mk (k E K C RG , m £M C RG ), and for K • RG denotes the right ideal generated by K in RG. * Research supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA T 16432 and 014279.
