International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 4, Pages 683-688
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.
In this paper, we generalize some well-known commutativity theorems for
associative rings as follows: Let , , , and be fixed non-negative integers such that
, or , and let be a ring with unity satisfying the polynomial identity
for all . Suppose that (i) has (that is implies
); (ii) the set of all nilpotent elements of is central for , and (iii) the set of
all zero-divisors of is also central for . Then is commutative. If is replaced by
and are relatively prime positive integers, then is commutative if extra constraint is
given. Other related commutativity results are also obtained.