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April, 2018 $b$-generalized $(\alpha,\beta)$-derivations and $b$-generalized $(\alpha,\beta)$-biderivations of Prime Rings
Vincenzo De Filippis, Feng Wei
Taiwanese J. Math. 22(2): 313-323 (April, 2018). DOI: 10.11650/tjm/170903

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.

Citation

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Vincenzo De Filippis. Feng Wei. "$b$-generalized $(\alpha,\beta)$-derivations and $b$-generalized $(\alpha,\beta)$-biderivations of Prime Rings." Taiwanese J. Math. 22 (2) 313 - 323, April, 2018. https://doi.org/10.11650/tjm/170903

Information

Received: 13 June 2017; Revised: 4 September 2017; Accepted: 14 September 2017; Published: April, 2018
First available in Project Euclid: 4 October 2017

zbMATH: 06965372
MathSciNet: MR3780719
Digital Object Identifier: 10.11650/tjm/170903

Subjects:
Primary: 16N60 , 16R50 , 16W25

Keywords: biderivation , generalized skew derivation , Prime ring

Rights: Copyright © 2018 The Mathematical Society of the Republic of China

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