Classification of Rings Satisfying Some Constraints on Subsets

Abstract. Let R be an associative ring with identity 1 and J(R) the Jacobson radical of R. Suppose that m ≥ 1 is a fixed positive integer and R an m-torsion-free ring with 1. In the present paper, it is shown that R is commutative if R satisfies both the conditions (i) [xm, ym] = 0 for all x, y ∈ R\J(R) and (ii) [x, [x, ym]] = 0, for all x, y ∈ R\J(R). This result is also valid if (ii) is replaced by (ii)’ [(yx)mxm − xm(xy)m, x] = 0, for all x, y ∈ R\N(R). Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).