Remarks on generalized Jordan $(\alpha, \beta)^\ast$-derivations of semiprime rings with involution
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Let $R$ be an associative ring with involution $\ast$ and $\alpha, \beta: R\rightarrow R$ ring homomorphisms. An additive mapping $d:R\rightarrow R$ is called an $(\alpha, \beta)^\ast$-derivation of $R$ if $d(xy)=d(x)\alpha(y^\ast)+\beta(x)d(y)$ is fulfilled for any $x,y \in R$, and an additive mapping $F:R\rightarrow R$ is called a generalized $(\alpha, \beta)^\ast$-derivation of $R$ associated with an $(\alpha, \beta)^\ast$-derivation $d$ if $F(xy)=F(x)\alpha(y^\ast)+\beta(x)d(y)$ is fulfilled for all $x,y \in R$. In this note, we intend to generalize a theorem of Vukman \cite{V}, and a theorem of Daif and El-Sayiad \cite{DS}, moreover, we generalize a theorem of Ali et al.~\cite{AFFK} and a theorem of Huang and Koc \cite{HK} related to generalized Jordan triple $(\alpha, \beta)^\ast$-derivations.
Abstract
Let $R$ be an associative ring with involution $\ast$ and $\alpha, \beta: R\rightarrow R$ ring homomorphisms. An additive mapping $d:R\rightarrow R$ is called an $(\alpha, \beta)^\ast$-derivation of $R$ if $d(xy)=d(x)\alpha(y^\ast)+\beta(x)d(y)$ is fulfilled for any $x,y \in R$, and an additive mapping $F:R\rightarrow R$ is called a generalized $(\alpha, \beta)^\ast$-derivation of $R$ associated with an $(\alpha, \beta)^\ast$-derivation $d$ if $F(xy)=F(x)\alpha(y^\ast)+\beta(x)d(y)$ is fulfilled for all $x,y \in R$. In this note, we intend to generalize a theorem of Vukman \cite{V}, and a theorem of Daif and El-Sayiad \cite{DS}, moreover, we generalize a theorem of Ali et al.~\cite{AFFK} and a theorem of Huang and Koc \cite{HK} related to generalized Jordan triple $(\alpha, \beta)^\ast$-derivations.
참고문헌 (14)
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