Rings and groups with commuting powers

Let n be a fixed positive integer. Let R be a ring with identity which satisfies (i) x n y n = y n x n 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 x k y k = y k x k and ( 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.