Two elementary commutativity theorems for generalized boolean rings

Abstract

In this paper we prove that if R is a ring with 1 as an identity element in which xm−xn∈Z(R) for all x∈R and fixed relatively prime positive integers m and n, one of which is even, then R is commutative. Also we prove that if R is a 2-torsion free ring with 1 in which (x2k)n+1−(x2k)n∈Z(R) for all x∈R and fixed positive integer n and non-negative integer k, then R is commutative.