{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T05:29:43Z","timestamp":1761802183144,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T00:00:00Z","timestamp":1761523200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11901030"],"award-info":[{"award-number":["11901030"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Natural Science Foundation of Beijing Municipality","award":["1232017"],"award-info":[{"award-number":["1232017"]}]},{"name":"Beijing Institute of Technology Research Fund Program for Young Scholars Open Research Fund of Hubei Key Laboratory of Mathematical Sciences","award":["MPL2024ORG004"],"award-info":[{"award-number":["MPL2024ORG004"]}]},{"name":"Youth Fund of Anhui Natural Science Foundation","award":["2008085QA01"],"award-info":[{"award-number":["2008085QA01"]}]},{"DOI":"10.13039\/501100009558","name":"Key Projects of the University Natural Science Research Project of Anhui Province","doi-asserted-by":"publisher","award":["KJ2019A0107"],"award-info":[{"award-number":["KJ2019A0107"]}],"id":[{"id":"10.13039\/501100009558","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work establishes a unified structural theory for non-global Lie higher derivations on triangular algebras T, without assuming the existence of a unit element. The primary contribution is the introduction of extreme non-global Lie higher derivations and the proof that every non-global Lie higher derivation on T admits a unique decomposition into three components: a higher derivation, an extreme non-global Lie higher derivation, and a central map vanishing on all commutators [x,y], where x,y\u2208T satisfy xy=0. This general framework is then explicitly applied to describe such derivations on two significant classes of algebras: upper triangular matrix algebras over faithful algebras and over semiprime algebras. By encompassing both unital and non-unital cases within a single characterization, the theory developed here not only generalizes numerous earlier results but also substantially expands the scope of the existing research landscape.<\/jats:p>","DOI":"10.3390\/axioms14110790","type":"journal-article","created":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T03:44:39Z","timestamp":1761795879000},"page":"790","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Non-Global Lie Higher Derivations on Triangular Algebras Without Assuming Unity"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6306-0578","authenticated-orcid":false,"given":"Xinfeng","family":"Liang","sequence":"first","affiliation":[{"name":"School of Mathematics and Big Data, AnHui University of Science & Technology, Huainan 232001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yujiao","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1080\/0308108031000096993","article-title":"Lie derivations of triangular algebras","volume":"51","author":"Cheung","year":"2003","journal-title":"Linear Multilinear Algebra"},{"key":"ref_2","first-page":"183","article-title":"Lie derivations of primitive rings","volume":"11","author":"Martindale","year":"1964","journal-title":"Linear Multilinear Algebra"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1215\/S0012-7094-73-04032-5","article-title":"Lie derivations of von Neumann algebras","volume":"40","author":"Miers","year":"1973","journal-title":"Duke Math. 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