On commutativity of rings with conditions involving elements and the Jacobson radical

Abstract

Let R be an associative ring with unity 1, N the set of nilpotents, J the Jacobson radical of R and n>1 be a fixed integer. We prove that if R is n(n+1)-torsion free and satisfies the identity (xy)ⁿ=yⁿxⁿ for all x,y ∈ R\ (N ∪ J), then R is commutative.