{
  "abstract": "Let $R$ be a commutative ring with nonzero identity and let $R^\\#$ denote the set of its nonzero nonunits. We extend the divisor topology $D(R)$, previously studied for integral domains, to arbitrary commutative rings and introduce the generalized divisor topology $GD(R)$ on $EC(R^\\#)$. Its basic open sets are \\[ B_a=\\{[b]\\in EC(R^\\#): b\\mid a^n \\text{ for some }n\\geq 1\\}. \\] The relation \\[ [b]\\in B_a \\quad\\Longleftrightarrow\\quad \\sqrt{aR}\\subseteq\\sqrt{bR} \\] shows that $GD(R)$ records radical divisibility among principal ideals. We prove that $GD(R)$ is an Alexandrov space and identify its Kolmogorov quotient with the poset of radicals of nonzero proper principal ideals. This description yields characterizations of the $T_0$ and discrete properties and of the equality $GD(R)=D(R)$. We also determine the isolated points of $GD(R)$. Further, we characterize nestedness, compactness, the Lindelöf property, and Noetherianity in terms of the order structure of radicals of principal ideals. In particular, for an integral domain $R$, $GD(R)$ is compact if and only if $R$ is a $G$-domain, while for a UFD the Lindelöf and Noetherian properties are determined by the number of nonassociate prime elements. Finally, we study the interaction of $GD(R)$ with multiplication and describe the behavior of its Kolmogorov quotient under surjective homomorphisms with nil kernel.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14867",
  "author_total": 4,
  "authors": [
    {
      "name": "Suat Koç"
    },
    {
      "name": "İrem Doğan"
    },
    {
      "name": "Dilara Erdemir"
    },
    {
      "name": "Ünsal Tekir"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e4czdbtjfbn87.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14867",
  "fulltext_url": null,
  "id": "e4czdbtjfbn87",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14867",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.GN"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7213361388",
  "pdf_url": "https://arxiv.org/pdf/2609.14867",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e4czdbtjfbn87.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14867",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14867"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Generalized divisor topology of commutative rings",
  "title_text": "Generalized divisor topology of commutative rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/e4czdbtjfbn87",
  "venue": "arXiv",
  "versions": []
}