International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 4, Pages 683-688
doi:10.1155/S0161171291000911
Commutativity theorems for rings with constraints on commutators
Department of Mathematics, Faculty of Science, King Abdul Aziz University, P. O. Box 31464, Jeddah 21497, Saudi Arabia
Received 29 August 1989; Revised 19 April 1991
Copyright © 1991 Hamza A. S. Abujabal. 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
In this paper, we generalize some well-known commutativity theorems for
associative rings as follows: Let n>1, m, s, and t be fixed non-negative integers such that
s≠m−1, or t≠n−1, and let R be a ring with unity 1 satisfying the polynomial identity
ys[xn,y]=[x,ym]xt for all y∈R. Suppose that (i) R has Q(n) (that is n[x,y]=0 implies
[x,y]=0); (ii) the set of all nilpotent elements of R is central for t>0, and (iii) the set of
all zero-divisors of R is also central for t>0. Then R is commutative. If Q(n) is replaced by
“m and n are relatively prime positive integers,” then R is commutative if extra constraint is
given. Other related commutativity results are also obtained.