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### $\left(\sigma ,\tau \right)$-derivations on prime near rings

Archivum Mathematicum

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example [1], [2], [3], [4], [5] and [8]) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like behaviour. In this paper we generalize some of the results due to Bell and Mason [4] on near-rings admitting a special type of derivation...

### $\left(\phi ,\varphi \right)$-derivations on semiprime rings and Banach algebras

Communications in Mathematics

Let $ℛ$ be a semiprime ring with unity $e$ and $\phi$, $\varphi$ be automorphisms of $ℛ$. In this paper it is shown that if $ℛ$ satisfies $2𝒟\left({x}^{n}\right)=𝒟\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)𝒟\left({x}^{n-1}\right)$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ is an ($\phi$, $\varphi$)-derivation. Moreover, this result makes it possible to prove that if $ℛ$ admits an additive mappings $𝒟,𝒢:ℛ\to ℛ$ satisfying the relations $\begin{array}{c}2𝒟\left({x}^{n}\right)=𝒟\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)𝒢\left(x\right)+𝒢\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)𝒢\left({x}^{n-1}\right)\phantom{\rule{0.166667em}{0ex}},\\ 2𝒢\left({x}^{n}\right)=𝒢\left({x}^{n-1}\right)\phi \left(x\right)+\varphi \left({x}^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left({x}^{n-1}\right)+\varphi \left(x\right)𝒟\left({x}^{n-1}\right)\phantom{\rule{0.166667em}{0ex}},\end{array}$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ and $𝒢$ are ($\phi$, $\varphi$)derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.

### A decomposition theorem for complete comodule algebras over complete Hopf algebras

Colloquium Mathematicae

### A Note About the Nowicki Conjecture on Weitzenböck Derivations

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 13N15, 13A50, 16W25.We reduce the Nowicki conjecture on Weitzenböck derivations of polynomial algebras to a well known problem of classical invariant theory.

### A note on a pair of derivations of semiprime rings.

International Journal of Mathematics and Mathematical Sciences

### A note on derivations in semiprime rings.

International Journal of Mathematics and Mathematical Sciences

### A Note on Inner Coalgebra Measuring and Derivations.

Mathematica Scandinavica

### A Note on Posner s Theorem with Generalized Derivations on Lie Ideals

Rendiconti del Seminario Matematico della Università di Padova

### A note on semiprime rings with derivation.

International Journal of Mathematics and Mathematical Sciences

### A Note on Skew Polynomial Rings

Publications de l'Institut Mathématique

### A note on the simplicity of skew polynomial rings of derivation type

Acta Mathematica Universitatis Ostraviensis

### A result on vanishing derivations for commutators on right ideals.

Mathematica Pannonica

### A theoreme on derivations in semiprime rings.

Collectanea Mathematica

### About relationship between generalized structurable algebras and Lie related triples.

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

### An identity with generalized derivations on Lie ideals, right ideals and Banach algebras

Czechoslovak Mathematical Journal

Let $R$ be a prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C=Z\left(U\right)$ the extended centroid of $R$, $L$ a non-central Lie ideal of $R$, $F$ a non-zero generalized derivation of $R$. Suppose that $\left[F\left(u\right),u\right]F\left(u\right)=0$ for all $u\in L$, then one of the following holds: (1) there exists $\alpha \in C$ such that $F\left(x\right)=\alpha x$ for all $x\in R$; (2) $R$ satisfies the standard identity ${s}_{4}$ and there exist $a\in U$ and $\alpha \in C$ such that $F\left(x\right)=ax+xa+\alpha x$ for all $x\in R$. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of...

### Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings

Czechoslovak Mathematical Journal

Let $R$ be a prime ring of characteristic different from 2 and 3, ${Q}_{r}$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n\ge 1$ a fixed positive integer. Let $\alpha$ be an automorphism of the ring $R$. An additive map $D:R\to R$ is called an $\alpha$-derivation (or a skew derivation) on $R$ if $D\left(xy\right)=D\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. An additive mapping $F:R\to R$ is called a generalized $\alpha$-derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F\left(xy\right)=F\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. We prove that, if $F$...

### Annihilators of skew derivations with Engel conditions on prime rings

Czechoslovak Mathematical Journal

Let $R$ be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring $Q$, $C$ the extended centroid of $R$ and $a\in R$. Suppose that $\delta$ is a nonzero $\sigma$-derivation of $R$ such that $a{\left[\delta \left({x}^{n}\right),{x}^{n}\right]}_{k}=0$ for all $x\in R$, where $\sigma$ is an automorphism of $R$, $n$ and $k$ are fixed positive integers. Then $a=0$.

### Applying the density theorem for derivations to range inclusion problems

Studia Mathematica

The problem of when derivations (and their powers) have the range in the Jacobson radical is considered. The proofs are based on the density theorem for derivations.

### Associated prime ideals of skew polynomial rings.

Beiträge zur Algebra und Geometrie

### Automorphisms and f-simplicity in skew polynomial rings

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

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