On commutativity of prime gamma rings with derivation

Let M be a weak Nobusawa Gamma-ring and gamma be a nonzero element of Gamma. In this paper, we find a relation between Gamma-rings and rings, and give some commutativity conditions on Gamma-rings by using this relation. Also, we prove that any Gamma-ring M in the sense of Nobusawa with a nonzero element gamma in the center of M-ring Gamma is gamma-prime if and only if M is Gamma-prime. As a consequence, we show that the semiprimeness (semisimpleness) of the ring (M, +, gamma) for any gamma is an element of Gamma implies the Gamma-semiprimeness(Gamma-semisimpleness) of the Gamma-ring M.