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2024 | OriginalPaper | Chapter

Generalized Derivations with Periodic Values on Prime Rings

Author : Giovanni Scudo

Published in: Advances in Ring Theory and Applications

Publisher: Springer Nature Switzerland

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Abstract

Let R be a prime ring whose characteristic is either zero or $p>0$ such that $p(2^n-2)$. Let $Q_r$ be its right Martindale quotient ring, C its extended centroid and F and G non-zero generalized derivations of R. If

$$ \biggl (F(x)x-xG(x)\biggr )^n=\biggl (F(x)x-xG(x)\biggr ) $$

for all $x\in [R,R]$, with $n\ge 2$ fixed integer, then one of the following holds:

there exists $a\in Q_r$ such that $F(x)=xa$ and $G(x)=ax$, for all $x\in R$;

$R\subseteq M_2(C)$, the ring of all $2\times 2$ matrices over C, and there exist $a,c\in R$ such that $F(x)=ax+xc$ and $G(x)=cx+xa$, for all $x\in R$;

$R\subseteq M_2(C)$, the ring of all $2\times 2$ matrices over C, and there exist $a,b,q\in R$ such that $F(x)=ax+xb$ and $G(x)=bx+xq$, for all $x\in R$, where $(a-q)^n=a-q$. Moreover C is periodic and, for all $x \in [R,R]$, $x^{2n}=x^2$.

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Title: Generalized Derivations with Periodic Values on Prime Rings
Author: Giovanni Scudo
Publisher: Springer Nature Switzerland
Book: Advances in Ring Theory and Applications
Print ISBN: 978-3-031-50794-6

Electronic ISBN: 978-3-031-50795-3

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-50795-3_17

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