Weakly Lie Engel condition on group rings

Abstract

Let R be a commutative ring of characteristic and G be a group. It is known that the group ring RG is bounded Lie Engel if and only if either G is nilpotent and G has a p-Abelian normal subgroup of finite p-power index (if n is a power of a prime p) or G is Abelian. In this paper we try to generalize this result; if x and y are elements of RG, let [x,y] = xy – yx and inductively, . Let m and n be two natural numbers. Among other results, we show that if RG satisfies , then either is a p-group (if n is a power of a prime p) or G is Abelian. If G is locally finite, then we show that RG satisfies if and only if either G is locally nilpotent and is a p-group (if n is a power of a prime p) or G is Abelian.