An extension of Herstein's theorem on Banach algebra
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 \mat...
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AIMS Press
2024-01-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2024201?viewType=HTML |
| _version_ | 1827358048089276416 |
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| author | Abu Zaid Ansari Suad Alrehaili Faiza Shujat |
| author_facet | Abu Zaid Ansari Suad Alrehaili Faiza Shujat |
| author_sort | Abu Zaid Ansari |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T05:57:07Z |
| format | Article |
| id | doaj.art-e2c208914f3e4a38877366c17474b0f0 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-08T05:57:07Z |
| publishDate | 2024-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-e2c208914f3e4a38877366c17474b0f02024-02-05T01:25:44ZengAIMS PressAIMS Mathematics2473-69882024-01-01924109411710.3934/math.2024201An extension of Herstein's theorem on Banach algebraAbu Zaid Ansari 0Suad Alrehaili1Faiza Shujat21. Department of Mathematics, Faculty of Science, Islamic University of Madinah, Saudi Arabia2. Department of Mathematics, Faculty of Science, Taibah University, Madinah, Saudi Arabia2. Department of Mathematics, Faculty of Science, Taibah University, Madinah, Saudi ArabiaLet $ \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.https://www.aimspress.com/article/doi/10.3934/math.2024201?viewType=HTMLsemiprime ringgeneralized left derivationalgebraic identitiesbanach algebra |
| spellingShingle | Abu Zaid Ansari Suad Alrehaili Faiza Shujat An extension of Herstein's theorem on Banach algebra AIMS Mathematics semiprime ring generalized left derivation algebraic identities banach algebra |
| title | An extension of Herstein's theorem on Banach algebra |
| title_full | An extension of Herstein's theorem on Banach algebra |
| title_fullStr | An extension of Herstein's theorem on Banach algebra |
| title_full_unstemmed | An extension of Herstein's theorem on Banach algebra |
| title_short | An extension of Herstein's theorem on Banach algebra |
| title_sort | extension of herstein s theorem on banach algebra |
| topic | semiprime ring generalized left derivation algebraic identities banach algebra |
| url | https://www.aimspress.com/article/doi/10.3934/math.2024201?viewType=HTML |
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