A note on the kernels of higher derivations

Jiantao Li; Xiankun Du

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Displaying similar documents to “A note on the kernels of higher derivations”

A note on linear derivations

Amit Patra (2024)

Czechoslovak Mathematical Journal

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At first we prove some results on a general polynomial derivation using few results of linear derivation. Then we study the ring of constants of a linear derivation for some rings. We know that any linear derivation is a nonsimple derivation. In the last section we find the smallest integer $w > 1$ such that the polynomial ring in $n$ variables is $w$ -differentially simple, all $w$ derivations are nonsimple and the $w$ derivations set contains a linear derivation.

On skew derivations as homomorphisms or anti-homomorphisms

Mohd Arif Raza, Nadeem ur Rehman, Shuliang Huang (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a prime ring with center $Z$ and $I$ be a nonzero ideal of $R$ . In this manuscript, we investigate the action of skew derivation $(δ, ϕ)$ of $R$ which acts as a homomorphism or an anti-homomorphism on $I$ . Moreover, we provide an example for semiprime case.

Generalized derivations on Lie ideals in prime rings

Basudeb Dhara, Sukhendu Kar, Sachhidananda Mondal (2015)

Czechoslovak Mathematical Journal

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Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$ . Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F (u) {[F (u), u]}^{n} = 0$ for all $u \in L$ , where $n \geq 1$ is a fixed integer. Then one of the following holds: (1) ...

Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings

Vincenzo de Filippis (2016)

Czechoslovak Mathematical Journal

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Let $R$ be a prime ring of characteristic different from 2, $Q_{r}$ 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 $α$ , and $p (x_{1}, ..., x_{n})$ is a non-central polynomial over $C$ such that $[F (x), α (y)] = G ([x, y])$ for all $x, y \in {p (r_{1}, ..., r_{n}) : r_{1}, ..., r_{n} \in R}$ . Then there exists $λ \in C$ such that $F (x) = G (x) = λ α (x)$ for all $x \in R$ .

On rings of constants of derivations in two variables in positive characteristic

Piotr Jędrzejewicz (2006)

Colloquium Mathematicae

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Let k be a field of chracteristic p > 0. We describe all derivations of the polynomial algebra k[x,y], homogeneous with respect to a given weight vector, in particular all monomial derivations, with the ring of constants of the form $k [x^{p}, y^{p}, f]$ , where $f \in k [x, y] ∖ k [x^{p}, y^{p}]$ .

Generalized derivations with power values on rings and Banach algebras

Abderrahman Hermas, Abdellah Mamouni, Lahcen Oukhtite (2024)

Mathematica Bohemica

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Let $R$ be a prime ring and $I$ a nonzero ideal of $R .$ The purpose of this paper is to classify generalized derivations of $R$ satisfying some algebraic identities with power values on $I .$ More precisely, we consider two generalized derivations $F$ and $H$ of $R$ satisfying one of the following identities: (1) ...

Generalized reverse derivations and commutativity of prime rings

Shuliang Huang (2019)

Communications in Mathematics

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Let $R$ be a prime ring with center $Z (R)$ and $I$ a nonzero right ideal of $R$ . Suppose that $R$ admits a generalized reverse derivation $(F, d)$ such that $d (Z (R)) \neq 0$ . In the present paper, we shall prove that if one of the following conditions holds: (i) $F (x y) \pm x y \in Z (R)$ , (ii) $F ([x, y]) \pm [F (x), y] \in Z (R)$ , (iii) $F ([x, y]) \pm [F (x), F (y)] \in Z (R)$ , (iv) $F (x \circ y) \pm F (x) \circ F (y) \in Z (R)$ , (v) $[F (x), y] \pm [x, F (y)] \in Z (R)$ , (vi) $F (x) \circ y \pm x \circ F (y) \in Z (R)$ for all $x, y \in I$ , then $R$ is commutative.