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Let M be a prime Γ-ring, I a nonzero ideal, θ an automorphism and dd a θ-derivation of MM. In this article we have proved the following result: (1) If d([x,y]α)=±([x,y]α) or d((xoy)α)=±((xoy)α) for x,y∈I;α∈Γ, then M is commutative. (2) Under the hypothesis dθ=θd and CharM ≠ 2, if (d(x) o d(y))α=0 or [d(x),d(y)]α=0 for all x,y∈I;α∈Γ, then M is commutative. (3) If dd acts as a homomorphism or an anti-homomorphism 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]