{
  "abstract": "Let p be a prime, let n≥2, and let R be a finite commutative local ring that is free of rank three over R0=Zpn and has residue field Fp. We show that R/pR is isomorphic either to Fp[t]/(t3) or to Fp[s,t]/(s,t)2, and that this dichotomy governs the ideal theory of R. In the first case, R≅R0[X]/(f) for a monic cubic polynomial f≡X3(modp); the ring R is a chain ring if p2∤f(0) and has embedding dimension two if p2∣f(0). In embedding dimension two, the nilpotency index t of the Jacobson radical J satisfies the sharp bounds n+2≤t≤2n+1, and we characterize the rings with J3=pJ2, for which all powers of J are given explicitly. In the second case, the embedding dimension is three, Ji=pi−1J for all i≥1, and t=n+1. In particular, no ring of embedding dimension two satisfies p∈J2. Using the left regular representation, we derive and solve the associativity equations for the structure constants and obtain an explicit six-parameter matrix model, together with matrix criteria for locality and for isomorphism. Finally, we show that R is a Frobenius ring if and only if its embedding dimension is at most two.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Sami H. Saif",
      "orcid": "0000-0002-6824-6985"
    },
    {
      "affs": [
        0
      ],
      "name": "Yousef Alkhamees",
      "orcid": "0000-0002-5779-0520"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9pag6wtcakzex.bib",
  "cited_by": 0,
  "doi": "10.3390/axioms15100755",
  "fulltext_url": null,
  "id": "9pag6wtcakzex",
  "issns": [
    "2075-1680"
  ],
  "landing_url": "https://www.mdpi.com/2075-1680/15/10/755",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7221338423",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/9pag6wtcakzex.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:30Z",
      "native_id": "10.3390/axioms15100755",
      "source": "crossref",
      "url": "https://www.mdpi.com/2075-1680/15/10/755"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Rings, Modules, and Algebras",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Finite Local Rings Free of Rank Three over Zpn: Radical Filtrations, Matrix Models, and the Frobenius Property",
  "title_text": "Finite Local Rings Free of Rank Three over Zpn: Radical Filtrations, Matrix Models, and the Frobenius Property",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:30Z",
  "url": "https://research.draconitic.com/paper/9pag6wtcakzex",
  "venue": "Axioms",
  "versions": []
}