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 79 Theorem 2.7. ([4], Theorem 2.7., page 87.) If G £ RAA 0 and R is a r ing with identity such that its additive group R + is torsion-free, then A U J(RG) = 0. 3. Residual Lie nilpotence It is clear, that A^(RG) = 0 if and only if G is an Abelian group. Therefore we may assume that the derived group G' = 72(C) of G is nontrivial. For a nilpotent group G the following inclusion is true (2) A^(RG) C A"{RG')RG (see in particular [4]). For every natural number i > 1 we define the normal subgroup Li = {g £ G'\g k £ 7i(G) for a suitable k > 1} of G. It is easy to see that 7 t(G) C L t and also that G /L x £ V 0 for every % > 1. An element g of a group G is called a generalized torsion element with respect to the lower central series of G if for every n the order of the elements 91n{G) of the factor group Gf^n{G) is finite. We recall that if the derived group G' of G contains no generalized torsion elements with respect to the lower central series of G, then G' has no generalized torsion elements with respect to the lower central series of G'. Theorem A. Let R be a commutative ring with identity, T(R + ) = 0 and let G' be with no generalized torsion elements with respect to the lower centra] series ofG. Then A^(RG) — 0 if and only if G is a residually-V 0 group. Proof. Since G' is with no generalized torsion elements with respect to the lower central series of G, then f~l = U) ^^ s o> G £ RPoConversely. Let G £ RP 0 and T(R + ) = 0. Since class V 0 is closed with respect to forming subgroups and finite direct products, by Lemmas 2.2. and 2.3. it is enough to show that A^(RG) = 0 for all G £ V 0. So let G £ V 0. Then by (2) A [u ]{RG) C A U(RG')RG. Because G' is a torsion-free nilpotent group, by Theorem 2.7. A i 0{RG l) = 0, and so, AM(RG) — 0. The proof is completed.
