Effect of Polynomial Identity x[x, y] = (x[x, y])n in the Commutativity of Rings

Abstract

. In this paper we study some sufficient conditions for commutativity of a ring according to Jacobsons’idea. Jacobson proved that if R is a ring satisfying x n = x (n > 1) for each x ∈ R, then R is commutative. In this paper, we show that R is commutative if for every x, y ∈ R there exists a positive integer n = n(x, y) such that (x[x, y])n = x[x, y]