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2024 | OriginalPaper | Chapter

Results on b-Generalized Derivations in Rings

Authors : Mohammad Salahuddin Khan, Shakir Ali, Abdul Nadim Khan, Mohammed Ayedh

Published in: Advances in Ring Theory and Applications

Publisher: Springer Nature Switzerland

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Abstract

This paper studies various identities of noncommutative prime rings involving b-generalized derivations. In particular, we prove that if a noncommutative prime ring \(\mathfrak {R}\) admits a nonzero b-generalized derivation \(\mathfrak {F}\) such that \([\mathfrak {F}(l), m]=[l, \mathfrak {F}(m)]\) for all \(l,m\in \mathfrak {I}\), where \(\mathfrak {I}\) is a nonzero ideal of \(\mathfrak {R}\), then \(\mathfrak {F}(l)=a l\) for all \(l\in \mathfrak {R}\) and for some \(a\in \mathcal {C}\) unless \(\mathfrak {R}\subseteq M_2(F)\), the \(2\times 2\) matrix ring over the field F. This gives a natural generalization of the results for derivations, generalized derivations and generalized \(\rho \)-derivations with an X-inner automorphism.

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previous chapter Generalized Skew-Derivations Acting on Multilinear Polynomials in Prime Rings

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

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Title: Results on b-Generalized Derivations in Rings
Authors: Mohammad Salahuddin Khan
Shakir Ali
Abdul Nadim Khan
Mohammed Ayedh
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_21

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