Rings and groups with commuting powers

Abstract

Letnbe a fixed positive integer. LetRbe a ring with identity which satisfies (i)xnyn=ynxnfor allx,yinR, and (ii) forx,yinR, there exists a positive integerk=k(x,y)depending onxandysuch thatxkyk=ykxkand(n,k)=1. ThenRis commutative. This result also holds for a groupG. It is further shown thatRandGneed not be commutative if any of the above conditions is dropped.