$b$-generalized $(\alpha,\beta)$-derivations and $b$-generalized $(\alpha,\beta)$-biderivations of Prime Rings

Abstract

Let $R$ be a ring, $\alpha$ and $\beta$ two automorphisms of $R$. An additive mapping $d \colon R \to R$ is called an $(\alpha,\beta)$-derivation if $d(xy) = d(x) \alpha(y) + \beta(x) d(y)$ for any $x,y \in R$. An additive mapping $G \colon R \to R$ is called a generalized $(\alpha,\beta)$-derivation if $G(xy) = G(x) \alpha(y) + \beta(x) d(y)$ for any $x,y \in R$, where $d$ is an $(\alpha,\beta)$-derivation of $R$. In this paper we introduce the definitions of $b$-generalized $(\alpha,\beta)$-derivation and $b$-generalized $(\alpha,\beta)$-biderivation. More precisely, let $d \colon R \to R$ and $G \colon R \to R$ be two additive mappings on $R$, $\alpha$ and $\beta$ automorphisms of $R$ and $b \in R$. $G$ is called a $b$-generalized $(\alpha,\beta)$-derivation of $R$, if $G(xy) = G(x) \alpha(y) + b\beta(x) d(y)$ for any $x,y \in R$.

Let now $D \colon R \times R \to R$ be a biadditive mapping. The biadditive mapping $\Delta \colon R \times R \to R$ is said to be a $b$-generalized $(\alpha,\beta)$-biderivation of $R$ if, for every $x,y,z \in R$, $\Delta(x,yz) = \Delta(x,y) \alpha(z) + b\beta(y) D(x,z)$ and $\Delta(xy,z) = \Delta(x,z) \alpha(y) + b\beta(x) D(y,z)$.

Here we describe the form of any $b$-generalized $(\alpha,\beta)$-biderivation of a prime ring.