Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)
KlRALY, B., The Lie augmentation terminals of group
64 Bertalan Király 2. Notations and some known facts. If H is a normal subgroup of G, then I(RH ) (or 1(H) for short when it is obvious from the context what ring R we are working with) denotes the ideal of RG generated by all elements of the form h — 1, (h £ II). It is well known that I (RH) is the kernel of the natural epimorphism 0 : RG —» RG /H induced by the group homomorphism <f> 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 E I), say, and ZF be its integral group ring (Z denotes the ring of rational integers). Then every homomorphism <j> : F —> G induces a ring homomorphism <f> : ZF —> RG by letting nyV ) — S ny4>(y)i where y £ F and the sum rungs over the finite set of n yy E ZF. If / E ZF, we denote by Aj(RG) the two-sided ideal of RG generated by the elements <f>(f ), cj> £ Hom(F,G), the set of homomorphism from F to G. In other words Aj(RG) is the ideal generated by the values of / in RG as the elements of G are substituted for the free generators X{-s. An ideal J of RG is called a polynomial ideal if J = Aj(RG) for some / £ ZF, F a free group. It is easy to see that the augmentation ideal A(RG) is a polynomial ideal. Really, A(RG) is generated as an R—module by the elements g — l(g £ G), i.e. by the values of the polynomial x — 1. From [3] (see also [2], Corollary 1.9, page 6) it follows the Lemma 2.1. ([2]) The Lie powers A^(RG),n > 1, are polynomial ideals in RG. We use also the following Lemma 2.2. ([2] Proposition 1.4, page 2) Let f £ ZF. Then f defines a polynomial ideal Aj(RG) in every group ring RG. Furter, if 6 : RG —> Ii II is a ring homomorphism induced by a group homomorphism <J> : G —» H and a ring homomorphism ip : R —• K , then 9(A f(RG)) C Aj(KH). (It is assumed here that IP(1R) = IK, where In and 1 K are identity of the rings R and K respectively.) Let (f) : RG —> RG / L be a natural epimorphism induced by the group homomorphism <p of G onto G/L. By Lemma 2.1 A^(RG)(n > 1) axe polynomial ideal and from Lemma 2.2 it follows that <f)(A^(RG)) = aW(RG/L). (1)
