On the Commutativity of a Ring with Identity

Abstract

AbstractLet R be a ring with identity. R satisfies one of the following properties for all x, y ∈ R:(I)xynxmy = xm+1yn+1 and mnm! n! x≠0 except x = 0;(II)xynxm = xm + 1yn + 1 and mm! n! x≠0 except x = 0;(III)xmyn = ynxm and m! n! x≠0 except x = 0;(IV)(xpyQ)n = xpnyqn for n = k, k + 1 and N(p, q, k) x≠0 except x = 0, where N(p, q, k) is a definite positive integer. Then R is commutative.