E®ect of Polynomial Identity x[x; y] = (x[x; y])n in the Commutativity of Rings

In this paper we study some su±cient conditions for com- mutativity of a ring according to Jacobsonsʹidea. Jacobson proved that if R is a ring satisfying xn = x (n > 1) for each x 2 R, then R is commutative. In this paper, we show that R is commutative if for every x; y 2 R there exists a positive integer n = n(x; y) such that (x[x; y])n = x[x; y].