Commutativity and structure of rings with commuting nilpotents

Hazar Abu-Khuzam,Adil Yaqub

doi:10.1155/s0161171283000101

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.

Highlights

Let R be a ring and let N denote the set of nilpotent elements of R
R is commutative and, 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

Summary

Introduction

Let R be a ring and let N denote the set of nilpotent elements of R. It is further shown that both conditions in hypothesis (iii) are essential. [i], the authors proved that if R is a semisimpIe ring with the property that, for all x,y in R there exists an integer n n(x,y) such that (xy) n n (yx) is in the center of R, R is commutative.

Results

Conclusion