Commutativity theorems for rings and groups with constraints on commutators

Abstract

Let n>1, m, t, s be any positive integers, and let R be an associative ring with identity. Suppose xt[xn,y]=[x,ym]ys for all x, y in R. If, further, R is n-torsion free, then R is commutativite. If n-torsion freeness of R is replaced by “m, n are relatively prime,” then R is still commutative. Moreover, example is given to show that the group theoretic analogue of this theorem is not true in general. However, it is true when t=s=0 and m=n+1.