International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 4, Pages 683-688

Commutativity theorems for rings with constraints on commutators

Hamza A. S. Abujabal

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.


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 sm1, or tn1, and let R be a ring with unity 1 satisfying the polynomial identity ys[xn,y]=[x,ym]xt for all yR. 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.