An extension of Herstein's theorem on Banach algebra

Abu Zaid Ansari; Suad Alrehaili; Faiza Shujat; Abu Zaid Ansari; Suad Alrehaili; Faiza Shujat

doi:10.3934/math.2024201

AIMS Mathematics

2024, Volume 9, Issue 2: 4109-4117. doi: 10.3934/math.2024201

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An extension of Herstein's theorem on Banach algebra

1.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Taibah University, Madinah, Saudi Arabia

Received: 04 October 2023 Revised: 24 November 2023 Accepted: 02 January 2024 Published: 12 January 2024
MSC : 16N60, 16B99, 16W25

Let $ \mathcal{A} $ be a $ (p+q)! $-torsion free semiprime ring. We proved that if $ \mathcal{H}, \mathcal{D} : \mathcal{A}\to \mathcal{A} $ are two additive mappings fulfilling the algebraic identity $ 2\mathcal{H}(a^{p+q}) = \mathcal{H}(a^p) a^q+ a^p \mathcal{D}(a^q)+\mathcal{H}(a^q) a^p+ a^q \mathcal{D}(a^p) $ for all $ a\in \mathcal{A} $, then $ \mathcal{H} $ is a generalized derivation with $ \mathcal{D} $ as an associated derivation on $ \mathcal{A} $. In addition to that, it is also proved in this article that $ \mathcal{H}_1 $ is a generalized left derivation associated with a left derivation $ \delta $ on $ \mathcal{A} $ if they fulfilled the algebraic identity $ 2\mathcal{H}_1(a^{p+q}) = a^p \mathcal{H}_1(a^q)+ a^q \delta(a^p)+a^q \mathcal{H}_1(a^p)+ a^p \delta(a^q) $ for all $ a \in \mathcal{A} $. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.
- semiprime ring,
- generalized left derivation,
- algebraic identities,
- Banach algebra
Citation: Abu Zaid Ansari, Suad Alrehaili, Faiza Shujat. An extension of Herstein's theorem on Banach algebra[J]. AIMS Mathematics, 2024, 9(2): 4109-4117. doi: 10.3934/math.2024201

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Abstract

Let $ \mathcal{A} $ be a $ (p+q)! $-torsion free semiprime ring. We proved that if $ \mathcal{H}, \mathcal{D} : \mathcal{A}\to \mathcal{A} $ are two additive mappings fulfilling the algebraic identity $ 2\mathcal{H}(a^{p+q}) = \mathcal{H}(a^p) a^q+ a^p \mathcal{D}(a^q)+\mathcal{H}(a^q) a^p+ a^q \mathcal{D}(a^p) $ for all $ a\in \mathcal{A} $, then $ \mathcal{H} $ is a generalized derivation with $ \mathcal{D} $ as an associated derivation on $ \mathcal{A} $. In addition to that, it is also proved in this article that $ \mathcal{H}_1 $ is a generalized left derivation associated with a left derivation $ \delta $ on $ \mathcal{A} $ if they fulfilled the algebraic identity $ 2\mathcal{H}_1(a^{p+q}) = a^p \mathcal{H}_1(a^q)+ a^q \delta(a^p)+a^q \mathcal{H}_1(a^p)+ a^p \delta(a^q) $ for all $ a \in \mathcal{A} $. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.

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