The commutativity in prime Gamma rings with left derivation

Abstract

Let M be a Γ− ring, U an ideal of M, charM � , 3 and 0 � d : M → M be a left derivation. In this paper we have proved the following results. (i) If d(U ) ⊂ U and d 2 (U) ⊂ Z then M is commutative where Z = {c ∈ M : cγm = mγc : m ∈ M, γ ∈ Γ} is the center of M. (ii) Let d1 ,d 2 be non zero left and right derivations of M and d2(U) ⊂ U. If d1d2(U) ⊂ Z then M is commutative.