{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:41:04Z","timestamp":1760146864118,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,17]],"date-time":"2024-12-17T00:00:00Z","timestamp":1734393600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["RSPD2024R871"],"award-info":[{"award-number":["RSPD2024R871"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper gives a thorough characterization of chain rings with index of nilpotency 5 and residue field Fpm, where p represents a prime number, contributing valuable insights to the field of algebraic structures. It carefully identifies and categorizes the family of chain rings with these specifications, thereby enhancing the understanding of their properties and applications. In addition, the work offers a detailed enumeration of all chain rings containing p5m elements. The significance of finite chain rings is emphasized, particularly in their suitability for coding theory, which confirms their relevance in contemporary mathematical and engineering contexts.<\/jats:p>","DOI":"10.3390\/axioms13120877","type":"journal-article","created":{"date-parts":[[2024,12,17]],"date-time":"2024-12-17T10:54:02Z","timestamp":1734432842000},"page":"877","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Commutative Chain Rings with Index of Nilpotency 5 and Residue Field Fpm"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-2520-2699","authenticated-orcid":false,"given":"Alhanouf Ali","family":"Alhomaidhi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6824-6985","authenticated-orcid":false,"given":"Sami","family":"Alabiad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9691-4979","authenticated-orcid":false,"given":"Nawal A.","family":"Alsarori","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1742","DOI":"10.3934\/math.2022100","article-title":"On classification of finite commutative chain rings","volume":"7","author":"Alabiad","year":"2021","journal-title":"AIMS Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1216\/rmjm\/1181072765","article-title":"A note of finite local rings","volume":"22","author":"Whelan","year":"1992","journal-title":"Rocky Mt. J. Math."},{"key":"ref_3","first-page":"810","article-title":"Finite rings with regular nilpotent graphs","volume":"12","author":"Kuzmina","year":"2015","journal-title":"Sib. \u00c8lektron. Mat. Izv."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01498378","article-title":"Algebraische Theorie der Ringe II","volume":"91","author":"Krull","year":"1924","journal-title":"Math. Ann."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/PL00012382","article-title":"On the structure of linear cyclic codes over finite chain rings","volume":"10","author":"Norton","year":"2000","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Alabiad, S., Alhomaidhi, A.A., and Alsarori, N.A. (2024). On linear codes over finite singleton local rings. 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On linear codes over local rings of order p4. Mathematics, 12.","DOI":"10.3390\/math12193069"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gassner, N., Greferath, M., Rosenthal, J., and Weger, V. (2022). Bounds for Coding Theory over Rings. Entropy, 24.","DOI":"10.3390\/e24101473"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3934\/amc.2022028","article-title":"Convolutional codes over finite chain rings, MDP codes and their characterization","volume":"17","author":"Alfarano","year":"2023","journal-title":"Adv. Math. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1353\/ajm.1999.0024","article-title":"Duality for modules over finite rings and applications to coding theory","volume":"121","author":"Wood","year":"1999","journal-title":"Am. J. Math."},{"key":"ref_14","first-page":"195","article-title":"Finite associative rings","volume":"21","author":"Raghavendran","year":"1969","journal-title":"Compos. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1006\/jabr.2000.8350","article-title":"Rings of order p5 Part II","volume":"231","author":"Corbas","year":"2000","journal-title":"Local Rings J. Algebra"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/0022-314X(72)90070-4","article-title":"On the group of units of certain rings","volume":"4","author":"Ayoub","year":"1972","journal-title":"J. Number Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"643","DOI":"10.2140\/pjm.1974.53.643","article-title":"Representations of finite rings","volume":"53","author":"Wilson","year":"1974","journal-title":"Pacific J. Math."},{"key":"ref_18","unstructured":"McDonald, B.R. (1974). Finite Rings with Identity, Marcel Dekker."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zariski, O., and Samuel, P. (1960). Commutative Algebra, Springer.","DOI":"10.1007\/978-3-662-29244-0"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Matsumura, H. (1986). Commutative Ring Theory, Cambridge University Press.","DOI":"10.1017\/CBO9781139171762"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/877\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:54:00Z","timestamp":1760115240000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/877"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,17]]},"references-count":20,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["axioms13120877"],"URL":"https:\/\/doi.org\/10.3390\/axioms13120877","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,12,17]]}}}