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Title: Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings (English)
Author: de Filippis, Vincenzo
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 66
Issue: 2
Year: 2016
Pages: 481-492
Summary lang: English
Category: math
Summary: Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n\geq 1$ a fixed positive integer. Let $\alpha $ be an automorphism of the ring $R$. An additive map $D\colon R\to R$ is called an $\alpha $-derivation (or a skew derivation) on $R$ if $D(xy)=D(x)y+\alpha (x)D(y)$ for all $x,y\in R$. An additive mapping $F\colon R\to R$ is called a generalized $\alpha $-derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F(xy)=F(x)y+\alpha (x)D(y)$ for all $x,y\in R$. We prove that, if $F$ is a nonzero generalized skew derivation of $R$ such that $F(x)\* [F(x),x]^n = 0$ for any $x\in L$, then either there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$, or $R\subseteq M_2(C)$ and there exist $a\in Q_r$ and $\lambda \in C$ such that $F(x)=ax+xa+\lambda x$ for any $x\in R$. (English)
Keyword: generalized skew derivation
Keyword: Lie ideal
Keyword: prime ring
MSC: 16N60
MSC: 16W25
idZBL: Zbl 06604481
idMR: MR3519616
DOI: 10.1007/s10587-016-0270-1
Date available: 2016-06-16T12:57:41Z
Last updated: 2018-07-02
Stable URL:
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