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|Paper IPM / M / 15380||
Let R be a commutative ring of characteristic n ≥ 0 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, [x, k y]=[[x, k−1 y],y]. Let m and n be two natural numbers. Among other results, we show that if RG satisfies [xm, n y]=0, then either G′ 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 [xm(x,y), n(x,y) y]=0 if and only if either G is locally nilpotent and G′ is a p-group (if n is a power of a prime p) or G is abelian.
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