On centrally-extended multiplicative (generalized)-$(\alpha,\beta)$-derivations in semiprime rings

Year 2020, Volume: 49 Issue: 2, 578 - 585, 02.04.2020

Najat Muthana Zakeiah Alkhamisi

https://doi.org/10.15672/hujms.568378

Cited By: 3

Abstract

Let $R$ be a ring with center $Z$ and $\alpha$, $\beta$ and $d$ mappings of $R$. A mapping $F$ of $R$ is called a centrally-extended multiplicative (generalized)-$(\alpha,\beta)$-derivation associated with $d$ if $F(xy)-F(x)\alpha(y)-\beta(x)d(y)\in Z$ for all $x, y \in R$. The objective of the present paper is to study the following conditions: (i) $F(xy)\pm \beta(x)G(y)\in Z$, (ii) $F(xy)\pm g(x)\alpha(y)\in Z$ and (iii) $F(xy)\pm g(y)\alpha(x)\in Z$ for all $x,y$ in some appropriate subsets of $R$, where $G$ is a multiplicative $($generalized$)$-$(\alpha,\beta)$-derivation of $R$ associated with the map $g$ on $R$.

Keywords

Semiprime ring, left ideal, multiplicative (generalized)-derivation, multiplicative (generalized)-$(\alpha;\beta)$-derivation, centrally-extended generalized $(\alpha;\beta)$-derivation, centrally-extended multiplicative (generalized)-$(\alpha;\beta)$-derivation, generalized $(\alpha;\beta)$-derivation

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