A commutativity property for rings

Abstract

We provide a partial answer to the following question: Assume that R is a finite ring of order s such that for every two subsets M and N of cardinalities m and n respectively, there exist x ∈ M and y ∈ N such that xy = yx. What relations among s, m, n guarantee that R is commutative? Also, we give criteria for commutativity of such rings in terms of structure and relations between m and n.