Rings and groups with commuting powers

Abstract

Let n be a fixed positive integer. Let R be a ring with identity which satisfies (i) xnyn=ynxn for all x,y in R, and (ii) for x,y in R, there exists a positive integer k=k(x,y) depending on x and y such that xkyk=ykxkand (n,k)=1. Then R is commutative. This result also holds for a group G. It is further shown that R and G need not be commutative if any of the above conditions is dropped.