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2024 | OriginalPaper | Chapter

Non-global Multiplicative Lie Triple Derivations on Rings

Authors : Mohammad Ashraf, Mohammad Afajal Ansari, Md Shamim Akhter

Published in: Advances in Ring Theory and Applications

Publisher: Springer Nature Switzerland

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Abstract

Let \(\mathfrak {R}\) be a ring containing a nontrivial idempotent with center \(\mathcal {Z}(\mathfrak {R})\). In the present article, it is shown that under certain restrictions every map \(\xi :\mathfrak {R}\rightarrow \mathfrak {R}\) (not necessarily additive) satisfying \(\xi ([[S, T], U])=[[\xi (S), T], U]+[[S,\xi (T)],\) \( U]+[[S, T],\xi (U)]\) for all \(S, T, U\in \mathfrak {R}\) with \(STU=0,\) is almost additive, that is, \(\xi (S+T)-\xi (S)-\xi (T)\in \mathcal {Z}(\mathfrak {R}).\) In addition, if \(\mathfrak {R}\) is a 2-torsion free prime ring, then \(\xi \) is of the form \(\xi =\partial +\eta ,\) where \(\partial \) is a derivation from \(\mathfrak {R}\) into its central closure \(\mathfrak {S}\) and \(\eta \) is a map from \(\mathfrak {R}\) into its extended centroid \(\mathfrak {C}\) such that \(\eta (S+T)-\eta (S)-\eta (T)\in \mathcal {Z}(\mathfrak {R})\) and \(\eta ([[S, T], U])=0\) for all \(S,T,U\in \mathfrak {R}\) with \(STU=0.\) The obtained results are then applied to standard operator algebras, factor von Neumann algebras and the algebra of all bounded linear operators.

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previous chapter The Noncommutative Singer-Wermer Conjecture and Generalized Skew Derivations

next chapter Algebraic Identities on Prime and Semiprime Rings

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Title: Non-global Multiplicative Lie Triple Derivations on Rings
Authors: Mohammad Ashraf
Mohammad Afajal Ansari
Md Shamim Akhter
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_15

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