International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 1, Pages 119-124
doi:10.1155/S0161171283000101
Commutativity and structure of rings with commuting nilpotents
1Department of Mathematics, University of Petroleum and Minerals, Dahran, Saudi Arabia
2Department of Mathematics, University of California, Santa Barbara 93106, California, USA
Received 28 September 1981
Copyright © 1983 Hazar Abu-Khuzam and Adil Yaqub. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.