Commutativity and structure of rings with commuting nilpotents

Abstract

Let R be a ring and let N denote the set of nilpotent elements of R. Let Z denote the center of R. Suppose that (i) N is commutative, (ii) for every x in R there exists x′ϵ<x> such that x−x2x′ϵN, where <x> denotes the subring generated by x, (iii) for every x,y in R, there exists an integer n=n(x,y)≥1 such that both (xy)n−(yx)n and (xy)n+1−(yx)n+1 belong to Z. Then R is commutative and, in fact, R is isomorphic to a subdirect sum of nil commutative rings and local commutative rings. It is further shown that both conditions in hypothesis (iii) are essential. The proof uses the structure theory of rings along with some earlier results of the authors.