{
  "abstract": "We introduce and study the so-termed {\\it $\\sqrtΔ$-fine rings}, a new class of rings that generalizes the classical {\\it fine rings} introduced by Călugăreanu-Lam in J. Algebra \\& Appl. (2016) by requiring that every nonzero element $r \\in R$ can be written as $r = u + a$, where $u$ is a unit and $a \\in \\sqrt{Δ(R)}$. We establish that every such ring is simple, every abelian $\\sqrtΔ$-fine ring is indecomposable, and most notably, the matrix ring $M_n(R)$ over a $\\sqrtΔ$-fine ring $R$ is again $\\sqrtΔ$-fine for every $n \\ge 1$. As a consequence, we characterize all semi-local $\\sqrtΔ$-fine rings as those rings which are precisely the simple Artinian rings. We also examine group rings, providing conditions under which they are either $\\sqrtΔ$-fine or generalized fine, where the latter class was introduced by Zhou in J. Algebra \\& Appl. (2022), and conclude our work with the difficult open question asking of whether each $\\sqrtΔ$-fine ring is necessarily fine.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30734",
  "author_total": 4,
  "authors": [
    {
      "name": "Peter Danchev"
    },
    {
      "name": "Omid Hasanzadeh"
    },
    {
      "name": "Ahmad Moussavi"
    },
    {
      "name": "Arash Javan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2wvkdy3agbjj3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30734",
  "fulltext_url": null,
  "id": "2wvkdy3agbjj3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30734",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7204952748",
  "pdf_url": "https://arxiv.org/pdf/2608.30734",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2wvkdy3agbjj3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2608.30734",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30734"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$\\sqrtΔ$-Fine Rings",
  "title_text": "√Δ-Fine Rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/2wvkdy3agbjj3",
  "venue": "arXiv",
  "versions": []
}