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2024 | OriginalPaper | Chapter

Commuting Iterates of Generalized Derivations on Lie Ideals

Author : Francesco Ammendolia

Published in: Advances in Ring Theory and Applications

Publisher: Springer Nature Switzerland

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Abstract

Let R be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring, C its extended centroid, F and G two nonzero generalized derivations of R, L a non-central Lie ideal of R and $n \ge 1$ a fixed integer. If

$$\begin{aligned} \Bigl \{F^2(x)x-xG^2(x)\Bigr \}^n=0 \end{aligned}$$

for all $x \in L$, then either $R \subseteq M_2(K)$, the ring $2 \times 2$ matrices over a field K, or one ot the following holds:

$F(x) = xa$ and $G(x) = xc$ for all $x \in R$ with $a^2 = c^2 \in C$;

$F(x) = xa$ and $G(x) = cx$ for all $x \in R$ with $a^2 = c^2$;

$F(x) = ax$ and $G(x) = xc$ for all $x \in R$ with $a^2 = c^2 \in C$;

$F(x) = ax$ and $G(x) = cx$ for all $x \in R$ with $a^2 = c^2 \in C$.

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previous chapter Results on Generalized Skew Bi-Semiderivation in Prime Rings

next chapter Non-linear Mappings Preserving Product on Factor von Neumann Algebras

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Title: Commuting Iterates of Generalized Derivations on Lie Ideals
Author: Francesco Ammendolia
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_23

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