{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T06:10:59Z","timestamp":1765260659351,"version":"3.46.0"},"reference-count":12,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T00:00:00Z","timestamp":1764288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007414","name":"Qassim University","doi-asserted-by":"crossref","award":["QU-APC-2025"],"award-info":[{"award-number":["QU-APC-2025"]}],"id":[{"id":"10.13039\/501100007414","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let R be an associative ring and M a left R-module. This paper examines the structural properties of the incidence module I(P,M), associated with a module M over a ring R and a locally finite poset P. We provide a complete characterization of when an additive derivation on I(P,M) is inner, for the case where P is a finite and connected poset. These criteria are then generalized to arbitrary posets, revealing a profound connection between the algebraic properties of the module and the graph-theoretic structure of P as a directed graph.<\/jats:p>","DOI":"10.3390\/axioms14120876","type":"journal-article","created":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T16:41:41Z","timestamp":1765212101000},"page":"876","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Additive Derivations of Incidence Modules"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-4586-8950","authenticated-orcid":false,"given":"Naseer","family":"Ullah","sequence":"first","affiliation":[{"name":"School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hailou","family":"Yao","sequence":"additional","affiliation":[{"name":"School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dalal","family":"Alhwikem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Imran Shabir","family":"Chuhan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Kotli AJK, Kotli 11100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,28]]},"reference":[{"key":"ref_1","unstructured":"Yang, Y., and Wei, F. (2021). Nonlinear Lie-Type derivations of finitary incidence algebras and related topics. arXiv."},{"key":"ref_2","unstructured":"Spiegel, E., and \u00d3 Donnel, C.J. (1997). Incidence Algebra, Marcel Dekker. Monographs and Textbooks, Pure and Applied Mathematics."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1090\/S0002-9939-1972-0313133-8","article-title":"Automorphisms and derivations of incidence algebras","volume":"36","author":"Baclawski","year":"1972","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2503","DOI":"10.1080\/00927872.2011.580441","article-title":"Derivations of finitary incidence rings","volume":"40","author":"Khripchenko","year":"2012","journal-title":"Commun. Algebra"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1236","DOI":"10.1090\/S0002-9904-1970-12617-9","article-title":"Structure of incidence algebras and their automorphism groups","volume":"76","author":"Stanley","year":"1970","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1357","DOI":"10.1216\/RMJ-2015-45-4-1357","article-title":"Jordan derivations of incidence algebras","volume":"45","author":"Xiao","year":"2015","journal-title":"Rocky Mt. J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1090\/S0002-9904-1961-10666-6","article-title":"Lie and Jordan structures in simple, associative rings","volume":"67","author":"Herstein","year":"1961","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1841","DOI":"10.1080\/00927872.2018.1523422","article-title":"Lie triple derivations of incidence algebras","volume":"47","author":"Wang","year":"2019","journal-title":"Commun. Algebra"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.laa.2016.10.011","article-title":"Lie derivations of incidence algebras","volume":"513","author":"Zhang","year":"2017","journal-title":"Linear Algebra Its Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bollob\u00e1s, B. (1998). Modern Graph Theory, Springer. Graduate Texts in Mathematics.","DOI":"10.1007\/978-1-4612-0619-4"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Bang-Jensen, J., and Gutin, G. (2009). Digraphs: Theory, Algorithms and Applications, Springer. [2nd ed.].","DOI":"10.1007\/978-1-84800-998-1"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"726","DOI":"10.1080\/00927872.2013.847951","article-title":"Multiplicative automorphisms of incidence algebras","volume":"43","author":"Brusamarello","year":"2015","journal-title":"Commun. Algebra"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/12\/876\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T05:10:44Z","timestamp":1765257044000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/12\/876"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,28]]},"references-count":12,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["axioms14120876"],"URL":"https:\/\/doi.org\/10.3390\/axioms14120876","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,11,28]]}}}