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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.

### 2-алгебры и тождества в них

Matematiceskie issledovanija

### A chain of Kurosh may have an arbitrary finite length

Czechoslovak Mathematical Journal

### A class of rings which are algebraic over the integers.

International Journal of Mathematics and Mathematical Sciences

### A concrete analysis of the radical concept.

Mathematica Pannonica

### A general concept of the pseudoprojective module

Czechoslovak Mathematical Journal

### A generalization of Mathieu subspaces to modules of associative algebras

Open Mathematics

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $𝒜$ , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to $𝒜$ -modules $ℳ$ . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable...

### A generalization of the prime radical of an ideal.

Divulgaciones Matemáticas

### A generalized Picard group for prime rings

Banach Center Publications

Semigroup forum

### A non-semiprime associative algebra with zero weak radical.

Extracta Mathematicae

The weak radical, W-Rad(A) of a non-associative algebra A, has been introduced by A. Rodríguez Palacios in [3] in order to generalize the Johnson's uniqueness of norm theorem to general complete normed non-associative algebras (see also [2] for another application of this notion). In [4], he showed that if A is a semiprime non-associative algebra with DCC on ideals, then W-Rad(A) = 0. In the first part of this paper we give an example of a non-semiprime associative algebra A with DCC on ideals and...

### 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 Radicals and Polynomial Rings.

Mathematica Scandinavica

### A note on rings which are multiplicatively generated by idempotents and nilpotents.

International Journal of Mathematics and Mathematical Sciences

### 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

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