Commutativity conditions on rings

Abstract

We prove the following result: let R be an arbitrary ring with centre Z such that for every x, y ∈ R, there exists a positive integer n = n(x, y) ≥ 1 such that (xy)ⁿ − yⁿxⁿ ∈ Z and (yx)ⁿ − xⁿyⁿ ∈ Z; then, if R has no non-zero nil ideals, R is commutative. We also prove a result on commutativity of general rings: if R is r!-torsion free and for all x, y ∈ R, [x^r, y^s] = 0 for fixed integers r ≥ s ≥ 1, then R is commutative. As a corollary we obtain that if R is (n + 1)!-torsion free and there exists a fixed n ≥ 1 such that (xy)ⁿ − yⁿxⁿ = (yx)ⁿ − xⁿyⁿ ∈ Z for all x, y ∈ R, then R is commutative.