The commutativity of prime Γ-rings with generalized skew derivations

Abstract

Let M be a prime Γ-ring with center {Z(M)}, and let θ be an automorphism of M. An additive map {d:M\to M} is called a skew derivation if {d(x\alpha y)=d(x)\alpha y+\theta(x)\alpha d(y)} for all {x,y\in M}, {\alpha\in\Gamma}. An additive map {F:M\to M} is called a generalized skew derivation if there exists a skew derivation {d:M\to M} such that {F(x\alpha y)=F(x)\alpha y+\theta(x)\alpha d(y)} holds for all {x,y\in M}, {\alpha\in\Gamma}. In the present paper, our main objective is to prove some commutativity results for prime Γ-rings M admitting a generalized skew derivation F satisfying anyone of the properties:(i){F(x\alpha y)\pm x\alpha y\in Z(M)},(ii){F(x\alpha y)\pm y\alpha x\in Z(M)},(iii){F(x)\alpha F(y)\pm x\alpha y\in Z(M)},(iv){F([x,y]_{\alpha})\pm[x,y]_{\alpha}=0},(v){F(\langle x,y\rangle_{\alpha})\pm\langle x,y\rangle_{\alpha}=0}for all {x,y\in I} and {\alpha\in\Gamma}. In fact, we obtain rather more general results which unify, extend and complement several well-known results proved in [3, 4, 5, 6, 32].