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### $\left(\phi ,\varphi \right)$-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 class of rings which are algebraic over the integers.

International Journal of Mathematics and Mathematical Sciences

### A generalized Picard group for prime rings

Banach Center Publications

Semigroup forum

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

International Journal of Mathematics and Mathematical Sciences

### A note on centralizers.

International Journal of Mathematics and Mathematical Sciences

### A note on derivations in semiprime rings.

International Journal of Mathematics and Mathematical Sciences

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

Rendiconti del Seminario Matematico della Università di Padova

### A note on rings with certain variable identities.

International Journal of Mathematics and Mathematical Sciences

### A Note on Semi-prime Rings.

Monatshefte für Mathematik

### A note on semiprime rings with derivation.

International Journal of Mathematics and Mathematical Sciences

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

Mathematica Pannonica

### A theoreme on derivations in semiprime rings.

Collectanea Mathematica

### Amalgamating commutative regular rings

Commentationes Mathematicae Universitatis Carolinae

### An identity on partial generalized automorphisms of prime rings

Rendiconti del Seminario Matematico della Università di Padova

### An identity related to centralizers in semiprime rings

Commentationes Mathematicae Universitatis Carolinae

The purpose of this paper is to prove the following result: Let $R$ be a $2$-torsion free semiprime ring and let $T:R\to R$ be an additive mapping, such that $2T\left({x}^{2}\right)=T\left(x\right)x+xT\left(x\right)$ holds for all $x\in R$. In this case $T$ is left and right centralizer.

### 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...

### Anneaux semi-premiers, noethériens, à identités polynômiales

Bulletin de la Société Mathématique de France

### 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 $F$...

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

Czechoslovak Mathematical Journal

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 $\alpha$, and $p\left({x}_{1},...,{x}_{n}\right)$ is a non-central polynomial over $C$ such that $\left[F\left(x\right),\alpha \left(y\right)\right]=G\left(\left[x,y\right]\right)$ for all $x,y\in \left\{p\left({r}_{1},...,{r}_{n}\right):{r}_{1},...,{r}_{n}\in R\right\}$. Then there exists $\lambda \in C$ such that $F\left(x\right)=G\left(x\right)=\lambda \alpha \left(x\right)$ for all $x\in R$.

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