On the Power Map and Ring Commutativity

Abstract

Let R denote an associative ring with 1, let n be a positive integer, and let k = 1, 2, or 3. The ring R will be called an (n, k)-ring if it satisfies the identitiesfor all integers m with n ≤ m ≤ n + k - 1. It was shown years ago by Herstein (See [2], [9], and [10]) that for n >1, any (n, l)-ring must have nil commutator ideal C(R). Later Luh [12] proved that primary (rc, 3)-rings must in fact be commutative, and Ligh and Richoux [11] recently showed that all (n, 3)-rings are commutative.