On Posner’s theorem with b-generalized skew derivations on Lie ideals

Let [Formula: see text] be a non-commutative prime ring of characteristic different from [Formula: see text] and [Formula: see text], [Formula: see text] its right Martindale quotient ring and [Formula: see text] its extended centroid. Suppose that [Formula: see text] is a non-central Lie ideal of [Formula: see text], [Formula: see text] a nonzero [Formula: see text]-generalized skew derivation of [Formula: see text]. If [Formula: see text] for all [Formula: see text], then one of the following holds: (a) there exists [Formula: see text] such that [Formula: see text], for all [Formula: see text]; (b) [Formula: see text], the ring of [Formula: see text] matrices over [Formula: see text], and there exist [Formula: see text] and [Formula: see text] such that [Formula: see text], for all [Formula: see text].

The commutativity of prime Γ-rings with generalized skew derivations

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].

Annihilators and Power Values of Generalized Skew Derivations on Lie Ideals

Let R be a prime ring of characteristic diòerent from 2, let Qr be its right Martindale quotient ring, and let C be its extended centroid. Suppose that F is a generalized skew derivation of R, L a non-central Lie ideal of and n, s ≥ 1 fixed integers. Iffor all u > L, then either R b Mz(C), the ring of 2 × 2 matrices over C, or m = 0 and there exists b ∊ Qr such that F(x) = bx, for any x ∊ R, with ab = 0.