{
  "abstract": "In this paper, introduce a new finiteness condition for commutative rings, called $Z$-Noetherian, under which every ideal contained in the set of zero-divisors $Z(R)$ is finitely generated. This condition lies properly between Noetherian and $Nil_{*}$-Noetherian. We establish intrinsic characterizations of $Z$-Noetherian rings and investigate their stability under nilpotent quotients and localizations. Under the assumption that $Z(R)$ is an ideal, we obtain a primary decomposition theorem for ideals contained in $Z(R)$. For $ZD$-rings, we prove a Cohen-type criterion characterizing $Z$-Noetherian in terms of the finite generation of prime ideals contained in $Z(R)$. We further establish $Z$-Noetherian analogues of the Hilbert Basis Theorem and Krull's Intersection Theorem. Finally, we establish a principal ideal theorem for $Z$-Noetherian rings. Collectively, these results show that several fundamental finiteness and structural properties of Noetherian rings extend to the broader class of $Z$-Noetherian rings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16178",
  "author_total": 3,
  "authors": [
    {
      "name": "Ambar Dubey"
    },
    {
      "name": "Tushar Singh"
    },
    {
      "name": "Shiv Datt Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/374xh6f4hn07k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16178",
  "fulltext_url": null,
  "id": "374xh6f4hn07k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16178",
  "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.AC"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7221807075",
  "pdf_url": "https://arxiv.org/pdf/2610.16178",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/374xh6f4hn07k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16178",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16178"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Z-Noetherian Rings",
  "title_text": "On Z-Noetherian Rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/374xh6f4hn07k",
  "venue": "arXiv",
  "versions": []
}