Commutativity theorems for rings with constraints on commutators

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.

Keywords

• commutativty of rings
• torsion free rings
• ring with unity
• semi-prime rings.