{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:46:11Z","timestamp":1760150771842,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,28]],"date-time":"2023-12-28T00:00:00Z","timestamp":1703721600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University","award":["PNURSP2023R87"],"award-info":[{"award-number":["PNURSP2023R87"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The study of symmetry is a fascinating and unifying subject that connects various areas of mathematics in the twenty-first century. Algebraic structures offer a framework for comprehending the symmetries of geometric objects in pure mathematics. This paper introduces new concepts in algebraic structures, concentrating on semimodules over semirings and analysing the neutrosophic structure in this context. We explore the properties of neutrosophic subsemimodules and neutrosophic ideals after defining them. We discuss, utilizing neutrosophic products, the representations of neutrosophic ideals and subsemimodules, as well as the relationship between neutrosophic products and intersections. Finally, we derive equivalent criteria in terms of neutrosophic structures for a semiring to be fully idempotent.<\/jats:p>","DOI":"10.3390\/sym16010041","type":"journal-article","created":{"date-parts":[[2023,12,28]],"date-time":"2023-12-28T09:35:21Z","timestamp":1703756121000},"page":"41","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Neutrosophic R-Structures in Semimodules over Semirings"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5596-5841","authenticated-orcid":false,"given":"Ghulam","family":"Muhiuddin","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Tabuk, P.O. Box-741, Tabuk 71491, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5529-0565","authenticated-orcid":false,"given":"Nabilah","family":"Abughazalah","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1414-2814","authenticated-orcid":false,"given":"Balasubramanian","family":"Elavarasan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore 641 114, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4097-3467","authenticated-orcid":false,"given":"Kasi","family":"Porselvi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore 641 114, India"}]},{"given":"Deena","family":"Al-Kadi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ahsan, J., Mordeson, J.N., and Shabir, M. (2012). Fuzzy Semirings with Applications to Automata Theory, Springer.","DOI":"10.1007\/978-3-642-27641-5"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"37","DOI":"10.14403\/jcms.2013.26.1.037","article-title":"On subtractive extension of subsemimodules of semimodules","volume":"26","author":"Chaudhari","year":"2013","journal-title":"J. Chungcheong Math. Soc."},{"key":"ref_3","first-page":"20","article-title":"On subsemimodules of semimodules","volume":"63","year":"2010","journal-title":"Bull. Acad. Stiinte Repub. Mold. Mat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2063","DOI":"10.1016\/j.jpaa.2017.08.020","article-title":"Valuations of semirings","volume":"222","author":"Jun","year":"2018","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Altassan, A., Mateen, M.H., and Pamucar, D. (2021). On Fundamental Theorems of Fuzzy Isomorphism of Fuzzy Subrings over a Certain Algebraic Product. Symmetry, 13.","DOI":"10.3390\/sym13060998"},{"key":"ref_7","first-page":"164","article-title":"Fuzzy bi-ideals in ternary semirings","volume":"3","author":"Kavikumar","year":"2009","journal-title":"Int. J. Comput. Math. Sci."},{"key":"ref_8","first-page":"151","article-title":"Solving dual fuzzy polynomial equation by ranking method","volume":"51","author":"Kavikumar","year":"2011","journal-title":"Far East J. Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Alsharari, F. (2021). On Single-Valued Neutrosophic Closure Spaces. Symmetry, 13.","DOI":"10.3390\/sym13081508"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_11","unstructured":"Smarandache, F. (1999). A Unifying Field in Logics, Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"583","DOI":"10.30948\/afmi.2017.14.6.583","article-title":"Neutrosophic \ud835\udd11-structures and their applications in semigroups","volume":"14","author":"Khan","year":"2017","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_13","first-page":"274","article-title":"Neutrosophic \ud835\udd11-ideals in semigroups","volume":"28","author":"Elavarasan","year":"2019","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Muhiuddin, G., Porselvi, K., Elavarasan, B., and Al-Kadi, D. (2022). Neutrosophic N-Structures in Ordered Semigroups. Comput. Model. Eng. Sci., 130.","DOI":"10.32604\/cmes.2022.018615"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Smarandache, F., and Abdel-Basset, M. (2021). The Determinant and Adjoint of an Interval-Valued Neutrosophic Matrix. Neutrosophic Operational Research, Springer.","DOI":"10.1007\/978-3-030-57197-9"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Jun, Y.B., Kim, S.J., and Smarandache, F. (2018). Interval neutrosophic sets with applications in BCK\/BCI-algebra. Axioms, 7.","DOI":"10.3390\/axioms7020023"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Muhiuddin, G., Ahmad, N., Al-Kenani Roh, E.H., and Jun, Y.B. (2019). Implicative neutrosophic quadruple BCK-algebras and ideals. Symmetry, 11.","DOI":"10.3390\/sym11020277"},{"key":"ref_18","first-page":"44","article-title":"Analysis of neutrosophic multiple regression","volume":"43","author":"Nagarajan","year":"2021","journal-title":"Neutrosophic Sets Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/41\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:43:24Z","timestamp":1760132604000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/41"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,28]]},"references-count":18,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["sym16010041"],"URL":"https:\/\/doi.org\/10.3390\/sym16010041","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,12,28]]}}}