A commutativity theorem for rings

Abstract

Let R be an associative ring with identity in which every element is either nilpotent or a unit. The following results are established. The set N of nilpotent elements in R is an ideal. If R/N is finite and if x ≡ y (mod N) implies x2 = y2 or both x and y commute with all elements of N, then R is commutative. Examples are given to show that R need not be commutative if “X2 = y2” is replaced by “xk = yK” for any integer k > 2. The case N = (0) yields Wedderburn's Theorem.