On the commutativity conditions for rings and Γ-rings

Abstract

Let R be any ring. In this paper we observe the relation between the center of R-ring R and the center of usual ring R and then prove if the center of R-ring R is nonzero, then R is commutative as a ring. We also show that the common hypothesisaαbβc = aβbαc for all a, b, c ∈ M and α, β ∈ Γfor a weak Nobusawa Γ-ring M is sufficent for M to be commutative. Also, we investigate some conditions on ideals of Γ-ring that make M to be commutative.