Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New Perspective
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> be a commutative ring with unity and <inline-formula&g...
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MDPI AG
2022-07-01
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| Online Access: | https://www.mdpi.com/2075-1680/11/7/328 |
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| author | Xinfeng Liang Dandan Ren Qingliu Li |
| author_facet | Xinfeng Liang Dandan Ren Qingliu Li |
| author_sort | Xinfeng Liang |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> be a commutative ring with unity and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">T</mi></semantics></math></inline-formula> be a triangular algebra over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>. Let a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mo>=</mo><msub><mrow><mo>{</mo><msub><mi>δ</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></msub></mrow></semantics></math></inline-formula> of nonlinear mappings <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mi>n</mi></msub><mo>:</mo><mi mathvariant="script">T</mi><mo>→</mo><mi mathvariant="script">T</mi></mrow></semantics></math></inline-formula> is a Lie triple higher derivation by local actions satisfying the equation. Under some mild conditions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">T</mi></semantics></math></inline-formula>, we prove in this paper that every Lie triple higher derivation by local actions on the triangular algebras is proper. As an application, we shall give a characterization of Lie triple higher derivations by local actions on upper triangular matrix algebras and nest algebras, respectively. |
| first_indexed | 2024-03-09T10:23:16Z |
| format | Article |
| id | doaj.art-79e2128d730c49a39df37292bc8eb900 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T10:23:16Z |
| publishDate | 2022-07-01 |
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| spelling | doaj.art-79e2128d730c49a39df37292bc8eb9002023-12-01T21:53:24ZengMDPI AGAxioms2075-16802022-07-0111732810.3390/axioms11070328Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New PerspectiveXinfeng Liang0Dandan Ren1Qingliu Li2School of Mathematics and Big Data, AnHui University of Science & Technology, Huainan 232001, ChinaSchool of Mathematics and Big Data, AnHui University of Science & Technology, Huainan 232001, ChinaSchool of Mathematics and Big Data, AnHui University of Science & Technology, Huainan 232001, ChinaLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula> be a commutative ring with unity and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">T</mi></semantics></math></inline-formula> be a triangular algebra over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>. Let a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mo>=</mo><msub><mrow><mo>{</mo><msub><mi>δ</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></msub></mrow></semantics></math></inline-formula> of nonlinear mappings <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mi>n</mi></msub><mo>:</mo><mi mathvariant="script">T</mi><mo>→</mo><mi mathvariant="script">T</mi></mrow></semantics></math></inline-formula> is a Lie triple higher derivation by local actions satisfying the equation. Under some mild conditions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">T</mi></semantics></math></inline-formula>, we prove in this paper that every Lie triple higher derivation by local actions on the triangular algebras is proper. As an application, we shall give a characterization of Lie triple higher derivations by local actions on upper triangular matrix algebras and nest algebras, respectively.https://www.mdpi.com/2075-1680/11/7/328Lie triple higher derivationfaithful bimodulehigher derivationlocal actiontriangular algebras |
| spellingShingle | Xinfeng Liang Dandan Ren Qingliu Li Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New Perspective Axioms Lie triple higher derivation faithful bimodule higher derivation local action triangular algebras |
| title | Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New Perspective |
| title_full | Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New Perspective |
| title_fullStr | Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New Perspective |
| title_full_unstemmed | Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New Perspective |
| title_short | Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New Perspective |
| title_sort | nonlinear lie triple higher derivations on triangular algebras by local actions a new perspective |
| topic | Lie triple higher derivation faithful bimodule higher derivation local action triangular algebras |
| url | https://www.mdpi.com/2075-1680/11/7/328 |
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