The commutativity of prime Γ-rings with generalized skew derivations

Abstract

Abstract Let M be a prime Γ-ring with center Z ⁢ ( M ) {Z(M)} , and let θ be an automorphism of M. An additive map d : M → M {d:M\to M} is called a skew derivation if d ⁢ ( x ⁢ α ⁢ y ) = d ⁢ ( x ) ⁢ α ⁢ y + θ ⁢ ( x ) ⁢ α ⁢ d ⁢ ( y ) {d(x\alpha y)=d(x)\alpha y+\theta(x)\alpha d(y)} for all x , y ∈ M {x,y\in M} , α ∈ Γ {\alpha\in\Gamma} . An additive map F : M → M {F:M\to M} is called a generalized skew derivation if there exists a skew derivation d : M → M {d:M\to M} such that F ⁢ ( x ⁢ α ⁢ y ) = F ⁢ ( x ) ⁢ α ⁢ y + θ ⁢ ( x ) ⁢ α ⁢ d ⁢ ( y ) {F(x\alpha y)=F(x)\alpha y+\theta(x)\alpha d(y)} holds for all x , y ∈ M {x,y\in M} , α ∈ Γ {\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 ⁢ α ⁢ y ) ± x ⁢ α ⁢ y ∈ Z ⁢ ( M ) {F(x\alpha y)\pm x\alpha y\in Z(M)} , (ii) F ⁢ ( x ⁢ α ⁢ y ) ± y ⁢ α ⁢ x ∈ Z ⁢ ( M ) {F(x\alpha y)\pm y\alpha x\in Z(M)} , (iii) F ⁢ ( x ) ⁢ α ⁢ F ⁢ ( y ) ± x ⁢ α ⁢ y ∈ Z ⁢ ( M ) {F(x)\alpha F(y)\pm x\alpha y\in Z(M)} , (iv) F ⁢ ( [ x , y ] α ) ± [ x , y ] α = 0 {F([x,y]_{\alpha})\pm[x,y]_{\alpha}=0} , (v) F ⁢ ( 〈 x , y 〉 α ) ± 〈 x , y 〉 α = 0 {F(\langle x,y\rangle_{\alpha})\pm\langle x,y\rangle_{\alpha}=0} for all x , y ∈ I {x,y\in I} and α ∈ Γ {\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].