On commutativity of completely prime gamma-rings

In this paper we prove that any completely prime g -ring m satisfying the condition aabb c = ab ba c (a,b,cîm anda,b îg) with nonzero derivation,is a commutative integral g -domain if its characteristic is not two. we also show that if the characteristic of m is 2 the g -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 g -ring m to the center of m when the characteristic of m is not two. if char m = 2,and aïz(m),then we show that the derivation is the inner derivation.