{
  "abstract": "Let $D$ be an integral domain and let $S\\subseteq D\\setminus\\{0\\}$ be multiplicative. We study the condition that, for each nonzero $a,b\\in D$, some $(a^n,b^n)_v$ is $S$-principal. A power estimate extends this binary condition to arbitrary finite families and permits synchronization of exponents. We describe the obstruction to descent from the AGCD domain $D_S$ by bounded $S$-torsion and by a finiteness condition on intersections of principal ideals. For an almost Prüfer $v$-multiplication domain, the almost $S$-GCD property, the AGCD property of $D_S$, and torsion of the $S$-class group are equivalent. In Krull domains this gives a criterion in terms of the divisor classes of height-one primes meeting $S$, and finite rational class rank reduces the multiplicative set to one element. We prove conductor transfer without a principal-contraction hypothesis, polynomial transfer in several settings, a normalization homeomorphism away from $S$, and a finite-principal-cover criterion with a Picard-group obstruction. Explicit examples distinguish localization, bounded descent, and the global and sublocal conditions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08340",
  "author_total": 3,
  "authors": [
    {
      "name": "Ahmed Hamed"
    },
    {
      "name": "Hwankoo Kim"
    },
    {
      "name": "Malik Tusif Ahmed"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a6drpvgb8f7kn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08340",
  "fulltext_url": null,
  "id": "a6drpvgb8f7kn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08340",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a6drpvgb8f7kn.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Almost S-GCD domains: localization, class groups, and extensions",
  "title_text": "Almost S-GCD domains: localization, class groups, and extensions",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:49Z",
  "url": "https://research.draconitic.com/paper/a6drpvgb8f7kn",
  "venue": "arXiv",
  "versions": []
}