Invariants of Skew Derivations

If is an automorphism and is a -derivation of a ring , then the subring of invariants is the set The main result of this paper is Theorem. Let be a -derivation of an algebra over a commutative ring such that

for all , where and .

(i)

If , then .

(ii)

If is a -stable left ideal of such that , then .

This theorem generalizes results on the invariants of automorphisms and derivations.

     

Generalized Skew Derivations with Power Central Values on Lie Ideals

Let R be a prime ring with center Z and L a noncommutative Lie ideal of R. Suppose that f is a right generalized β-derivation of R associated with a β-derivation δ such that f(x) ⁿ  ∈ Z for all x ∈ L, where n is a fixed positive integer. Then f = 0 unless dim  _C RC = 4.     

Central Extensions and Derivations of the Lie Algebras of Skew Derivations for the Quantum Torus

ABSTRACT

In this article, we study the central extensions and derivations of the Lie algebra of skew derivations for the quantum torus. The results of the article generalize those obtained in Jiang and Meng (1998a Jiang , C. , Meng , D. ( 1998a ). The derivation algebra of the associative algebra C _q [X, Y, X ^?1, Y ^?1] . Comm. Algebra 26 : 1723 – 1736 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] b Jiang , C. , Meng , D. ( 1998b ). The automorphism group of the derivation algebra of the Virasoro-like algebra . Adv. Math. (China) 27 : 175 – 183 . [CSA]  [Google Scholar]) and Kirkman et al. (1994 Kirkman , E. , Procesi , C. , Small , L. ( 1994 ). A q-analog for the Virasoro algebra . Comm. Algebra 22 : 3755 – 3774 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

On Rings with Locally Nilpotent Skew Derivations

In this article, we examine algebras with a locally nilpotent q-skew σ-derivation d when there is an element x such that d(x) = 1 and either q is not a root of 1 or q = 1 in characteristic zero. When characteristic p > 0, we also examine the situation where d is an ordinary derivation.     

环的广义斜导子

设R是一个半素环, RF(resp．Q)是它的左Martindale商环(对称Martindale 商环),K是R的一个本质理想,则K上的每一个广义斜导子μ能被唯一地扩展到RF和Q 上．设R是一个素环,K是R的一个本质理想,μ是K上的一个广义斜导子且α为其伴随自同构,d为其伴随斜导子,如果存在n≥0,使得对任意的x∈K都有μ(x)n=0,那么μ=0．     

Derivations and Skew Polynomial Rings

Soient D un corps non nécessairement commutatif et L un sous-corps de D. On établit une condition nécessaire et suffisante pour que le groupe multiplicatif L de L soit d'indice fini dans son normalisateur N dans D. Lorsque la dimension à gauche [D : L]_g est un nombre premier, on précise le groupe N/L et la structure de D.     

Skew Derivations of Prime Rings

Given a prime ring R, a skew g-derivation for g : R → R is an additive map f : R → R such that f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x)) = g(f(x)) for all x, y ∈ R. We generalize some properties of prime rings with derivations to the class of prime rings with skew derivations.     

Skew Derivations and Engel Conditions

It is known that for a nonzero derivation d of a prime ring R, if a nonzero ideal I of R satisfies the Engel-type identity [[…[[d(x ^{k ₀} ), x ^{k ₁} ], x ^{k ₂} ],…], x ^{k _n} ], then R is commutative. Here we extend this result to a skew derivation of R for a Lie ideal I, which has an immediate corollary that replaces d by an automorphism of R. A related result in two variables is obtained for d a (θ, ?)-derivation.     

On Skew Derivations in Semiprime Rings

We investigate the commutativity in a (semi-)prime ring R which admits skew derivations δ ₁, δ ₂ satisfying [δ ₁(x), δ ₂(y)]?=?[x, y] for all x, y in a nonzero right ideal of R. This result is a natural generalization of Bell and Daif’s theorem on strong commutativity preserving derivations and a recent result by Ali and Huang.     

Generalized Skew Derivations Cocentralizing Multilinear Polynomials

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. Suppose that F, G are generalized skew derivations of R and \({f(x_1, \ldots, x_n)}\) is a non-central multilinear polynomial over C with n non-commuting variables. If F and G satisfy the following condition:

$$F(f(r_1,\ldots, r_n))f(r_1, \ldots,r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))\in C$$

for all \({r_1, \ldots, r_n \in R}\), then we describe all possible forms of F and G.

     

Generalized Skew Derivations with Nilpotent Values in Prime Rings

Let ? be a prime ring, 𝒞 the extended centroid of ?, ? a Lie ideal of ?, F be a nonzero generalized skew derivation of ? with associated automorphism α, and n ≥ 1 be a fixed integer. If (F(xy) ? yx) ⁿ  = 0 for all x, y ∈ ?, then ? is commutative and one of the following statements holds:

(1) Either ? is central;

(2) Or ? ? M ₂(𝒞), the 2 × 2 matrix ring over 𝒞, with char(𝒞) = 2.     

Generalized Derivations with Skew Nilpotent Values on Lie Ideals

This article gives characterizations of generalized derivations with skew nilpotent values on noncommutative Lie ideals of a prime ring. The results simultaneously generalize the ones of Herstein, Lee and Carini et al.     

Power-Commuting Generalized Skew Derivations in Prime Rings

Let R be a non-commutative prime ring of characteristic different from 2 with extended centroid C, F ≠ 0 a generalized skew derivation of R, and n ≥ 1 such that [F(x), x] ⁿ  = 0, for all x ∈ R. Then there exists an element λ ∈ C such that F(x) = λx, for all x ∈ R.     

Generalized Skew Derivations Implemented by Elementary Operators

Let R be a semiprime ring with the maximal right ring of quotients Q _mr . An additive map d: R →Q _mr is called a generalized skew derivation if there exists a ring endomorphism σ:R →R and a map \(\d:R \to Q_{mr}\) such that \(d(xy)=\d(x)y+\sigma(x)d(y)\) for all x,y?∈?R. If σ is surjective, we determine the structure of generalized skew derivations for which there exists a finite number of elements a _i ,b _i ?∈?Q _mr such that d(x)?=?a ₁ xb ₁?+???+?a _n xb _n for all x?∈?R.     

Annihilators of Skew Derivations with Engel Conditions on Lie Ideals

Let R be a prime ring, L a noncentral Lie ideal of R, and a ∈ R. Set [x, y]₁ = [x, y] = xy ? yx for x, y ∈ R and inductively [x, y]_k = [[x, y]_k?1, y] for k > 1. Suppose that δ is a nonzero σ-derivation of R such that a[δ(x), x]_k = 0 for all x ∈ L, where σ is an automorphism of R and k is a fixed positive integer. Then a = 0 except when char R = 2 and R ? M₂(F), the 2 × 2 matrix ring over a field F.