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

76 Bertalan Király For the first limit ordinal u we adopt the notation: oo A

[uj

](RG) = P|i4M(£G). i=i The ideal A(RG

) of the group ring RG is said to be residually Lie nilpotent if Alwine) = 0. 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. Our main results are given in section 3. These results (Theorem A, B and C) are rather technical so they are not stated in the introduction. 2. Notations and some known facts If H is a normal subgroup of G

, then I (RH) (or 1(H) for short) denotes the ideal of RG generated by elements of the form h — 1, (h £ H). It is well known that I (RH) is the kernel of the natural epimorphism <f)\ RG RG / H induced by the group homomorphism 0 of G onto G

/H. It is clear that I(RG) = A(RG). Let F be a free group on the free generators X{ (i £ I) and ZF be its integral group ring (Z denotes the ring of rational integers). Then every homomorphism (f>: F —> G induces a ring homomorphism <j>\ ZF —» RG by letting (f)(Y^ n

yy) = Yy

ny

(f

>(y)- If / £ ZF

> w

e denote by Af(RG) the twosided ideal of RG generated by the elements </>(/), (f) £ Hom(F, G), the set of homomorphism from F to G. In other words Af(RG) is the ideal generated by the values of / in RG as the elements of G are substituted for the free generators s. An ideal J of RG is called a polynomial ideal if J = Af(RG) for some / £ ZF. It is easy to see that the augmentation ideal A(RG) is a polynomial ideal. Really, A(RG) is generated as an Ä-module by elements g — 1 (g £ G), i.e. by the values of the polynomial x — 1. We also use the following Lemma 2.1. ([4], Proposition 1.4., page 2.) Let f £ ZF. Then f defines a polynomial ideal Aj(RG) in every group ring RG. Further, if 9: RG —> KH