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
78 Bertalan Király Aío — the class of torsion-free nilpotent groups. Afp — the class of nilpotent p-groups of bounded exponent. AÍq = U p€f iA/" p and T*n = U p enV p, where Q is a subset of the set of primes. The ideal J p(R) of a ring R is defined by J p(R) = fl™ = lp nR. Theorem 2.4. ([4], Theorem 2.13., page 85.) Let G be a residually V p-group and J p(R) = 0. Then A^(RG) = 0. We shall use the following lemma, which gives some elementary properties of the Lie powers of A(RG). Lemma 2.5. ([4], Proposition 1.7., page 4.) For arbitrary natural numbers n and m are true: (1) /(t„(G))C AM(RG), (2) [AM(RG),AW(RG)] C Al n+ m\RG), (3) AW(RG)>AW(RG) C A^ n+T n-^(RG), where 7 n(G) is the nth term of the lower central series of G. We write D^\(RG) for the nth Lie dimension subgroup D^(RG) of G over R. That is D [n ](RG) = {g G G\g- 1 e A^(RG)}. By Lemma 2.5. it follows that for every natural number n the inclusion ln(G) C D [n ](RG) holds . We also use the following theorems Theorem 2.6. ([1], Theorem 3.2.) Let a group G contain a nontrivial generalized torsion element. Then A(RG) is residually nilpotent if and only if there exists a non-empty subset 0 of the set of primes such that Hpefi J P(R) = 0, G is discriminated by the class Mu and for every proper subset A of the set Q at least one of the conditions (1) n pe A/p(Ä) = 0 (2) G is discriminated by the. class of groups A/ft \ a holds. Let T(i? +) denote the torsion subgroup of the additive group of a ring R and let A U(RG) = n°l 1A n(RG), where A n(RG) is the nth associative power of A(RG).
