{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T11:40:08Z","timestamp":1759146008756,"version":"3.44.0"},"reference-count":10,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T00:00:00Z","timestamp":1759017600000},"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":"publisher","award":["12271481"],"award-info":[{"award-number":["12271481"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we provide the necessary and sufficient conditions for a Lie centralizer to be proper and a Jordan centralizer to be a centralizer on graded rings. Since every Lie centralizer naturally induces an anti-symmetric mapping and every Jordan centralizer naturally induces a symmetric mapping, our results provide the underlying graded structures reflected in these mappings. As applications, we recover known results for trivial extension algebras and triangular algebras, and additionally characterize Lie (Jordan) centralizers on exterior algebras, whose operations are inherently anti-symmetric.<\/jats:p>","DOI":"10.3390\/sym17101611","type":"journal-article","created":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T10:49:34Z","timestamp":1759142974000},"page":"1611","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lie (Jordan) Centralizer on Graded Rings"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-0699-3260","authenticated-orcid":false,"given":"Haoting","family":"He","sequence":"first","affiliation":[{"name":"School of Mathematics Sciences, Zhejiang University of Technology, Hangzhou 310023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qikai","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics Sciences, Zhejiang University of Technology, Hangzhou 310023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haiyan","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Mathematics Sciences, Zhejiang University of Technology, Hangzhou 310023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1080\/00927872.2020.1797759","article-title":"Lie (Jordan) centralizers on generalized matrix algebras","volume":"49","author":"Jabeen","year":"2021","journal-title":"Commun. 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Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2375","DOI":"10.1080\/03081087.2022.2104788","article-title":"Characterizations of Lie centralizers of triangular algebras","volume":"71","author":"Liu","year":"2023","journal-title":"Linear Multilinear Algebra"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1122","DOI":"10.1016\/j.laa.2011.07.014","article-title":"Semi-centralizing maps of generalized matrix algebras","volume":"436","author":"Li","year":"2012","journal-title":"Linear Algebra Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1007\/s10998-018-0260-1","article-title":"k-commuting mappings of generalized matrix algebras","volume":"79","author":"Li","year":"2019","journal-title":"Period. Math. Hung."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Hazrat, R. (2016). Graded Rings and Graded Grothendieck Groups, Cambridge University Press.","DOI":"10.1017\/CBO9781316717134"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2693","DOI":"10.1080\/03081087.2020.1810605","article-title":"On nonlinear Lie centralizers of generalized matrix algebras","volume":"70","author":"Liu","year":"2022","journal-title":"Linear Multilinear Algebra"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1016\/j.laa.2010.09.015","article-title":"Additivity of multiplicative maps on triangular rings","volume":"434","author":"Wang","year":"2011","journal-title":"Linear Algebra Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Darling, R.W.R. (1994). Differential Forms and Connections, Cambridge University Press.","DOI":"10.1017\/CBO9780511805110"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/10\/1611\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T11:21:43Z","timestamp":1759144903000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/10\/1611"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,28]]},"references-count":10,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["sym17101611"],"URL":"https:\/\/doi.org\/10.3390\/sym17101611","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,28]]}}}