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 $R$.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez:Let $R$ be a ring with center $Z$ and $\alpha$, $\beta$ and $d$ mappings of $R$.
📊
Temel Bulgular & Veri: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 $x, y \in R$.
💡
Sonuç & Katkı: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 $R$.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Semiprime ring · left ideal · multiplicative (generalized)-derivation
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 $R$.
Muthana, N., Alkhamisi, Z. (2022). On centrally-extended multiplicative (generalized)-$(\alpha,\beta)$-derivations in semiprime rings. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 2, 578–585. https://ajindex.com/articles/529649
A. Ali, B. Dhara, S. Khan and F. Ali, Multiplicative (generalized)-derivations and left ideals in semiprime rings, Hacettepe J. Math. Stat. 44 (6), 1293–1306, 2015
H.E. Bell and M.N. Daif, On centrally-extended maps on rings, Beitrage Algebra Geom. Article No. 244, 1–8, 2015
B. Dhara and S. Ali, On multiplicative (generalized)-derivations in prime and semiprime rings, Aequat. Math. 86 (1-2), 65–79, 2013
C. Lanski, An Engel condition with derivation for left ideals, Proc. Amer. Math. Soc. 125 (2), 339–345, 1997
M.S. Tammam El-Sayiad, N.M. Muthana and Z.S. Alkhamisi, On rings with some kinds of centrally-extended maps, Beitr¨age Algebra Geom. Article No. 274, 1–10, 2015.