{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:42:42Z","timestamp":1760146962337,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T00:00:00Z","timestamp":1735257600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Researchers Supporting Project","award":["RSPD2024R871"],"award-info":[{"award-number":["RSPD2024R871"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper presents a comprehensive characterization of finite local rings of length 4 and with residue field Fpm, where p is a prime number. Such rings have an order of p4m elements. The current paper provides the structure and classification, up to isomorphism, of local rings consisting of p4m elements. We also give the exact number of non-isomorphic classes of these rings with fixed invariants p,n,m,k. In particular, we have listed all finite local rings of 4-length and of order p8 and 256.<\/jats:p>","DOI":"10.3390\/axioms14010012","type":"journal-article","created":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T09:13:32Z","timestamp":1735290812000},"page":"12","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Finite Local Rings of Length 4"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6824-6985","authenticated-orcid":false,"given":"Sami","family":"Alabiad","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-2520-2699","authenticated-orcid":false,"given":"Alhanouf Ali","family":"Alhomaidhi","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\/0000-0002-9691-4979","authenticated-orcid":false,"given":"Nawal A.","family":"Alsarori","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,27]]},"reference":[{"key":"ref_1","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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Commun."},{"key":"ref_13","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 Appl."},{"key":"ref_14","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_15","unstructured":"McDonald, B.R. (1974). Finite Rings with Identity, Marcel Dekker."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Zariski, O., and Samuel, P. (1960). Commutative Algebra, Springer.","DOI":"10.1007\/978-3-662-29244-0"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Matsumura, H. (1986). Commutative Ring Theory, Cambridge University Press.","DOI":"10.1017\/CBO9781139171762"},{"key":"ref_18","first-page":"195","article-title":"Finite associative rings","volume":"21","author":"Raghavendran","year":"1969","journal-title":"Compos. Math."},{"key":"ref_19","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_20","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. 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