On Commutativity of Completely Prime Gamma-Rings

In this paper we prove that any completely prime Gamma-ring M satisfying the condition alpha alpha b beta c = alpha beta b alpha c (a, b, c is an element of M and alpha, beta is an element of Gamma) with nonzero derivation, is a commutative integral Gamma-domain if its characteristic is not two. We also show that if the characteristic of M is 2 the Gamma-ring M is either commutative or is an order in a simple 4-dimensional algebra over its center. We give necessary condition in terms of derivations for belongings of an element of the Gamma-ring M to the center of M when the characteristic of M is not two. If char M = 2, and a is an element of Z(M), then we show that the derivation is the inner derivation.