On commutativity of rings with some polynomial constraints

Abstract

Let R be an associative ring with unity 1, N(R) the set of nilpotents, J(R) the Jacobson radical of R and n > 1 be a fixed integer. We prove that R is commutative if and only if it satisfies (xy)n = ynxn for all x, y ∈ R \ N(R) and commutators in R are n(n + 1)-torsion free. Moreover, we extend the same result in the case when x, y ∈ R\J(R).