On the commutativity conditions for rings and Γ-rings

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 hypothesis a alpha b beta c = a beta b alpha c for all a, b, c. M and a, beta is an element of Gamma for a weak Nobusawa Gamma-ring M is sufficent for M to be commutative. Also, we investigate some conditions on ideals of Gamma-ring that make M to be commutative.