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2024 | OriginalPaper | Chapter

Commutativity Theorems on Prime Rings with Generalized Derivations

Authors : Basudeb Dhara, Sukhendu Kar, Kalyan Singh

Published in: Advances in Ring Theory and Applications

Publisher: Springer Nature Switzerland

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Abstract

Suppose that $\mathcal {R}$ is a prime ring, $\mathcal {I}$ a nonzero ideal of $\mathcal {R}$, $\mathcal {F}$ a generalized derivation of $\mathcal {R}$ and n a fixed positive integer. If

$$ (\mathcal {F}(x_1)x_2+x_1\mathcal {F}(x_2)+\mathcal {F}(x_2)x_1+x_2\mathcal {F}(x_1))^n-(x_1x_2+x_2x_1)=0, $$

for all $x_1,x_2\in \mathcal {I}$, then one of the following holds:

$\mathcal {R}$ is commutative;

$n=1$ and there exists $\lambda \in \mathcal {C}$ such that $\mathcal {F}(x)=\lambda x$, for all $x\in \mathcal {R}$ with $2\lambda =1$.

If $char(\mathcal {R})\ne 2$ and

$$ (\mathcal {F}(x_1)x_2+x_1\mathcal {F}(x_2)+\mathcal {F}(x_2)x_1+x_2\mathcal {F}(x_1))^n-(x_1x_2+x_2x_1)\in \mathcal {Z}(\mathcal {R}), $$

for all $x_1,x_2\in \mathcal {I}$, then one of the following holds:

$\mathcal {R}$ is commutative.

$n=1$ and there exists $\lambda \in \mathcal {C}$ such that $\mathcal {F}(x)=\lambda x$, for all $x\in \mathcal {R}$ with $2\lambda =1$.

We examine the aforementioned identities in semiprime rings and also obtain some range inclusion results on Banach algebras.

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previous chapter Specht Modules and Representations of Symmetric Group

next chapter Commutativity of -Prime Rings with Generalized Derivation

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Title: Commutativity Theorems on Prime Rings with Generalized Derivations
Authors: Basudeb Dhara
Sukhendu Kar
Kalyan Singh
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_5

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