{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:10:39Z","timestamp":1760058639099,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T00:00:00Z","timestamp":1744675200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["RSPD2025R871"],"award-info":[{"award-number":["RSPD2025R871"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study investigates finite commutative local non-chain rings characterized by the well-established invariants p,\u00a0n,\u00a0m,\u00a0l, and k, where p denotes a prime number. We specifically focus on Frobenius local rings with length l=5 and an index of nilpotency t=4. The significance of Frobenius rings in coding theory arises when specific results of linear codes are applicable to both finite fields and finite Frobenius rings. In light of this, we provide a comprehensive classification and enumeration of Frobenius local rings of order p5m with t=4, highlighting their distinctive properties in relation to varying values of n. This research advances our understanding of the structural features of Frobenius rings and their applications in coding theory.<\/jats:p>","DOI":"10.3390\/axioms14040296","type":"journal-article","created":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T03:59:06Z","timestamp":1744689546000},"page":"296","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Class of Local Non-Chain Rings of Order p5m"],"prefix":"10.3390","volume":"14","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"}]},{"given":"Badriyah Rashed","family":"Alshahrani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"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"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,15]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","unstructured":"Zariski, O., and Samuel, P. (1960). Commutative Algebra, Springer.","DOI":"10.1007\/978-3-662-29244-0"},{"key":"ref_3","first-page":"195","article-title":"Finite associative rings","volume":"21","author":"Raghavendran","year":"1969","journal-title":"Compos. Math."},{"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":"1","DOI":"10.1016\/j.ffa.2016.08.004","article-title":"Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3","volume":"43","year":"2017","journal-title":"Finite Fields Their Appl."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","unstructured":"Alabiad, S., Alhomaidhi, A.A., and Alsarori, N.A. (2024). MacWilliams identities and generator matrices for linear codes over Zp4[u]\u2329u2\u2212p3\u03b2,pu\u232a. Axioms, 13.","DOI":"10.3390\/axioms13080552"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s00200-019-00384-0","article-title":"On codes over Frobenius rings: Generating characters, MacWilliams identities and generator matrices","volume":"30","author":"Dougherty","year":"2019","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Alabiad, S., Alhomaidhi, A.A., and Alsarori, N.A. (2024). On linear codes over local rings of order p4. Mathematics, 12.","DOI":"10.3390\/math12193069"},{"key":"ref_10","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_11","unstructured":"Wirt, B.R. (1972). Finite Non-Commutative Local Rings. [Ph.D. Thesis, University of Oklahoma]."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1006\/jabr.2000.8350","article-title":"Rings of order p5 Part II. Local Rings","volume":"231","author":"Corbas","year":"2000","journal-title":"J. Algebra"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Matsumura, H. (1986). Commutative Ring Theory, Cambridge University Press.","DOI":"10.1017\/CBO9781139171762"},{"key":"ref_14","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":"Pac. J. Math."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Alkhamees, Y., and Alabiad, S. (2022). The structure of local rings with singleton basis and their enumeration. Mathematics, 10.","DOI":"10.3390\/math10214040"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1007\/PL00000451","article-title":"Characterization of finite Frobenius rings","volume":"76","author":"Honold","year":"2001","journal-title":"Arch. Math."},{"key":"ref_17","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_18","unstructured":"McDonald, A.R. (1974). Finite Rings with Identity, Marcel Dekker."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2350034","DOI":"10.1142\/S0129167X23500349","article-title":"Expanders on Matrices over a Finite Chain Ring, I","volume":"34","author":"Ha","year":"2023","journal-title":"Int. J. Math."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Alabiad, S., Alhomaidhi, A.A., and Alsarori, N.A. (2024). On linear codes over finite singleton local rings. Mathematics, 12.","DOI":"10.3390\/math12071099"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/296\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:14:38Z","timestamp":1760030078000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/296"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,15]]},"references-count":20,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040296"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040296","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,4,15]]}}}