- Hacettepe Journal of Mathematics and Statistics
- Vol: 49 Issue: 2
- On centrally-extended multiplicative (generalized)-$(\alpha,\beta)$-derivations in semiprime rings
On centrally-extended multiplicative (generalized)-$(\alpha,\beta)$-derivations in semiprime rings
Authors : Najat Muthana, Zakeiah Alkhamisi
Pages : 578-585
Doi:10.15672/hujms.568378
View : 12 | Download : 4
Publication Date : 2020-04-02
Article Type : Research
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)-$(alpha, eta)$-derivation, centrally-extended generalized $(alpha, eta)$-derivation, centrally-extended multiplicative (generalized)-$(alpha, eta)$-derivation, gen