{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,4]],"date-time":"2026-08-04T19:11:28Z","timestamp":1785870688724,"version":"3.56.0"},"reference-count":13,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,10,24]],"date-time":"2025-10-24T00:00:00Z","timestamp":1761264000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia","award":["RGP.2\/340\/46"],"award-info":[{"award-number":["RGP.2\/340\/46"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper presents the novel framework of neutrosophic modules, an algebraic structure that arises by superimposing neutrosophic sets on classical module theory. The core of this study lies in the investigation of the structural symmetry between the axioms of a module and the trivalent nature of the neutrosophic set. We define a new class of modules based on the neutrosophic set. In addition, this study establishes and examines the categorical structure corresponding to neutrosophic R-modules. Furthermore, it presents the procedures by which finitely generated variants of these modules can be formulated and studied. Partial characterizations are given for a particular type in which the distribution of neutrosophic values remains within a finite set.<\/jats:p>","DOI":"10.3390\/sym17111797","type":"journal-article","created":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T02:50:48Z","timestamp":1761533448000},"page":"1797","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Finitely Generated Neutrosophic Modules with Finite Value Distribution"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6267-6255","authenticated-orcid":false,"given":"Amr","family":"Elrawy","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8516-0868","authenticated-orcid":false,"given":"Ali Yahya","family":"Hummdi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61471, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,24]]},"reference":[{"key":"ref_1","unstructured":"Smarandache, F. (1998). Neutrosophy, Neutrosophic Probability, Set and Logic: Analytic Synthesis & Synthetic Analysis, American Research Press."},{"key":"ref_2","unstructured":"Smarandache, F. (1999). A Unifying Field in Logics: Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press."},{"key":"ref_3","unstructured":"Kandasamy Vasantha, W.B., and Smarandache, F. (2004). Basic Neutrosophic Algebraic Structures and Their Application to Fuzzy and Neutrosophic Models, Pro Quest Information & Learning."},{"key":"ref_4","first-page":"287","article-title":"Neutrosophic set\u2014A generalization of the intuitionistic fuzzy set","volume":"24","author":"Smarandache","year":"2005","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_5","unstructured":"Smarandache, F. (2017). 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Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"35964","DOI":"10.3934\/math.20241705","article-title":"Neutrosophic modules over modules","volume":"9","author":"Hummdi","year":"2024","journal-title":"AIMS Math."},{"key":"ref_11","unstructured":"Mac Lane, S. (1998). Categories for the Working Mathematician, Springer Science & Business Media."},{"key":"ref_12","unstructured":"Lam, T.Y. (2012). Lectures on Modules and Rings, Springer Science & Business Media."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Rotman, J.J. (2009). An Introduction to Homological Algebra, Springer.","DOI":"10.1007\/b98977"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1797\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T03:08:06Z","timestamp":1761534486000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1797"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,24]]},"references-count":13,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["sym17111797"],"URL":"https:\/\/doi.org\/10.3390\/sym17111797","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,24]]}}}