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International Journal of Mathematics and Mathematical Sciences
Volume 4, Issue 1, Pages 101-107
http://dx.doi.org/10.1155/S0161171281000069

Rings and groups with commuting powers

DEPARTMENT OF MATHEMATICS, PETROLEUM UNIVERSITY, Saudi Arabia

Received 5 March 1979

Copyright © 1981 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.