ON COMMUTATIVITY OF PRIME Γ-RINGS WITH θ-DERIVATIONS.

Let M be a prime Γ-ring, I a nonzero ideal, θ an automorphism and d a θ-derivation of M. In this article we have proved the following result: (1) If d([x, y]α) = ±([x, y]α) or d((x o y)α) = ±((x o y)α) for all x, y ε I;α ε Γ, then M is commutative. (2) Under the hypothesis dθ = θd and Char M, 2, if (d(x) o d(y))α = 0 or [d(x),d(y)]α = 0 for all x, y 2 I;α α Γ, then M is commutative. (3) If d acts as a homomorphism or an antihomomorphism on I, then d = 0 or M is commutative. Moreover, an example is given to demonstrate that the primeness imposed on the hypothesis of the various results is essential. [ABSTRACT FROM AUTHOR]