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].

m-potent commutators involving skew derivations and multilinear polynomials

Let {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1<m\in\mathbb{Z}^{+}}, {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over {\mathscr{C}} and δ is a skew derivation of {\mathscr{R}}.

On graded derivations of superalgebras

In this paper, we find conditions under which the bracket defined by a graded derivation on a Lie superalgebra (g, [, ]) is skew-supersymmetry and satisfies the super Jacobi identity, so it defines the structure of a Lie superalgebra on g.In the case of the algebra of differential forms on a supermanifold, we study the graded commutator of graded derivations, graded skew-derivations and a graded derivation, with another graded skew-derivation of the superalgebra of differential forms on a supermanifold.

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.

Skew Derivations with Power Central Values on Commutators

Let R be a prime ring with center [Formula: see text], δ: R → R a nonzero skew derivation, and n a fixed positive integer. In this paper, we show that R is a commutative ring if (i) [δ([x,y]),[x,y]]n = 0 for all x, y ∈ R or (ii) [Formula: see text] for all x ∈ R, except some specific cases.

GK DIMENSION AND LOCALLY NILPOTENT SKEW DERIVATIONS

Let A be a domain over an algebraically closed field with Gelfand–Kirillov dimension in the interval [2,3). We prove that if A has two locally nilpotent skew derivations satisfying some natural conditions, then A must be one of five algebras. All five algebras are Noetherian, finitely generated, and have Gelfand–Kirillov dimension equal to 2. We also obtain some results comparing the Gelfand–Kirillov dimension of an algebra to its subring of invariants under a locally nilpotent skew derivation.

DERIVATIONS AND SKEW DERIVATIONS OF THE GRASSMANN ALGEBRAS

Surprisingly, skew derivations rather than ordinary derivations are more basic (important) object in study of the Grassmann algebras. Let Λn = K ⌊x1, …, xn⌋ be the Grassmann algebra over a commutative ring K with ½ ∈ K, and δ be a skew K-derivation of Λn. It is proved that δ is a unique sum δ = δ ev + δ od of an even and odd skew derivation. Explicit formulae are given for δev and δod via the elements δ (x1), …, δ (xn). It is proved that the set of all even skew derivations of Λn coincides with the set of all the inner skew derivations. Similar results are proved for derivations of Λn. In particular, Der K(Λn) is a faithful but not simple Aut K(Λn)-module (where K is reduced and n ≥ 2). All differential and skew differential ideals of Λn are found. It is proved that the set of generic normal elements of Λn that are not units forms a single Aut K(Λn)-orbit (namely, Aut K(Λn)x1) if n is even and two orbits (namely, Aut K(Λn)x1 and Aut K(Λn)(x1 + x2 ⋯ xn)) if n is odd.