Skew Derivations Research Articles

A Note on n-Commuting Generalized Skew Derivations on Prime Rings

A Note on n-Commuting Generalized Skew Derivations on Prime Rings

Annihilators of Power Central Values of Generalized Skew Derivations on Lie Ideals

Annihilators of Power Central Values of Generalized Skew Derivations on Lie Ideals

Generalized Skew Derivations with Hypercommuting Conditions on Lie Ideals

Generalized Skew Derivations with Hypercommuting Conditions on Lie Ideals

Skew derivations of incidence algebras

Skew derivations of incidence algebras

B-Generalized Skew Derivations on Multilinear Polynomials

b-Generalized Skew Derivations on Multilinear Polynomials

Central Values Involving Generalized Skew Derivations and Polynomials in Prime Rings

Central Values Involving Generalized Skew Derivations and Polynomials in Prime Rings

Classical Radicals of Invariants of Algebraic Skew Derivations

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

Skew derivations on partially ordered sets

Skew derivations on partially ordered sets

Power-Central Values and Engel Conditions in Prime Rings with Generalized Skew Derivations

Let R be a prime ring of characteristic different from 2 with extended centroid C, $$n\ge 1$$ a fixed positive integer, $$F, G:R\rightarrow R$$ two non-zero generalized skew derivations of R.

Power Values of Generalized Skew Derivations with Annihilator Conditions on Lie Ideals

Let R be a prime ring with maximal right ring of quotients Q, extended centroid C and a noncommutative Lie ideal L. Our aim is to characterize the forms of generalized skew derivations $$\delta $$ and $$\varrho $$ of R in case of $$\begin{aligned} p(\delta (u)^{l_{1}}\varrho (u)^{l_{2}}\delta (u)^{l_{3}}\varrho (u)^{l_{4}}\cdots \varrho (u)^{l_{k}} )^{n}=0, \end{aligned}$$ for all $$u\in L$$ , where $$0 \ne p\in R$$ , $$l_{1},l_{2},\ldots ,l_{k}$$ are fixed nonnegative integers with $$l_{1}\ne 0$$ and n is a positive fixed integer. The proof of this main result is based on the so-called extended Jacobson density theorem, which is different from that of previous results in the literature. Also as a result, we obtain if $$p\delta (u)^{n}=0$$ for all $$u \in L$$ , then there exists $$a\in Q$$ such that $$\delta (x)=ax$$ for all $$x \in R$$ and $$pa=0$$ unless R satisfies $$s_4$$ .

Generalized Skew Derivations and Nilpotent Values on Lie Ideals

Let R be a prime ring of characteristic different from 2 and 3, Qr be its right Martindale quotient ring and C be its extended centroid. Suppose that F and G are generalized skew derivations of R, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Under appropriate conditions we prove that if (F(x)x – xG(x))n = 0 for all x ∈ L, then one of the following holds: (a) there exists c ∈ Qr such that F(x) = xc and G(x) = cx; (b) R satisfies s4 and there exist a, b, c ∈ Qr such that F(x) = ax + xc, G(x) = cx + xb and (a − b)2 = 0.

English

English

B-Generalized Skew Derivations on Lie Ideals

Let R be a non-commutative prime ring, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, $$F\ne 0$$ an b-generalized skew derivation of R, L a non-central Lie ideal of R, $$0\ne a\in R$$ and $$n\ge 1$$ a fixed integer. In this paper, we prove the following two results:

An Engel condition with b-generalized derivations for Lie ideals

Let [Formula: see text] be a prime ring with the extended centroid [Formula: see text], [Formula: see text] a noncommutative Lie ideal of [Formula: see text] and [Formula: see text] a nonzero [Formula: see text]-generalized derivation of [Formula: see text]. For [Formula: see text], let [Formula: see text]. We prove that if [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers, then there exists [Formula: see text] such that [Formula: see text] for all [Formula: see text] except when [Formula: see text], the [Formula: see text] matrix ring over a field [Formula: see text]. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75–80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139–152.]

Centralizers of X-Generalized Skew Derivations on Multilinear Polynomials in Prime Rings

Let R be a prime ring of characteristic different from 2, $$Q_r$$ be its right Martindale quotient ring and C be its extended centroid, G be a nonzero X-generalized skew derivation of R, and S be the set of the evaluations of a multilinear polynomial $$f(x_1,\ldots ,x_n)$$ over C with n non-commuting variables. Let $$u,v \in R$$ be such that $$uG(x)x+G(x)xv=0$$ for all $$x\in S$$ . Then one of the following statements holds:

Skew derivations on generalized Weyl algebras

A wide class of skew derivations on degree-one generalized Weyl algebras R(a,φ) over a ring R is constructed. All these derivations are twisted by a degree-counting extensions of automorphisms of R. It is determined which of the constructed derivations are Q-skew derivations. The compatibility of these skew derivations with the natural Z-grading of R(a,φ) is studied. Additional classes of skew derivations are constructed for generalized Weyl algebras given by an automorphism φ of a finite order. Conditions that the central element a that forms part of the structure of R(a,φ) needs to satisfy for the orthogonality of pairs of aforementioned skew derivations are derived. In addition local nilpotency of constructed derivations is studied. General constructions are illustrated by description of all skew derivations (twisted by a degree-counting extension of the identity automorphism) of generalized Weyl algebras over the polynomial ring in one variable and with a linear polynomial as the central element.

Strong Commutativity Preserving Skew Derivations in Semiprime Rings

Let R be a semiprime ring of characteristic different from 2, C its extended centroid, Z(R) its center, F and G non-zero skew derivations of R with associated automorphism \(\alpha \) and m, n positive integers such that $$\begin{aligned}{}[F(x),G(y)]_m=[x,y]^n ~\mathrm{for \, all}~x,y \in R. \end{aligned}$$ Then R is commutative.

Annihilators and Power Values of Generalized Skew Derivations on Lie Ideals

AbstractLet 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.

Identities with Product of Generalized Skew Derivations on Multilinear Polynomials

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, and C its extended centroid. Suppose that F, G are generalized skew derivations of R, with the same associated automorphism, and f(x1,…, xn) a noncentral multilinear polynomial over C with n noncommuting variables, such thatfor all r1,…, rn ∈ R. Then we describe all possible forms of F and G.

Engel-Type Conditions Involving Two Generalized Skew Derivations in Prime Rings

Let ℛ be a prime ring of characteristic different from 2, 𝒬r the right Martindale quotient ring of ℛ, 𝒞 the extended centroid of ℛ, F, G two generalized skew derivations of ℛ, and k ≥ 1 be a fixed integer. If [F(r), r]kr − r[G(r), r]k = 0 for all r ∈ ℛ, then there exist a ∈ 𝒬r and λ ∈ 𝒞 such that F(x) = xa and G(x) = (a + λ)x, for all x ∈ ℛ.