{
  "abstract": "A multiplicative set $S$ of a commutative ring $R$ is strongly multiplicative if every family $(s_i)_{i \\in I}$ of elements of $S$ admits a common multiple in $S \\cap \\bigcap_{i \\in I} s_iR$. We combine the structural results on strongly multiplicative sets with their prime-theoretic and module-theoretic applications. We prove that $S$ is strongly multiplicative if and only if localization at $S$ commutes with arbitrary intersections of ideals, if and only if $R_S$ is the localization at an idempotent, and if and only if $D(S)$ is clopen in $\\Spec(R)$. We then develop permanence results under homomorphisms, products, factor rings, trivial extensions, and amalgamations; describe the associated split torsion theory; show that almost multiplicative sets contribute no new cases beyond their multiplicative hull; and record a Mittag--Leffler refinement for intersections of submodules in finitely generated modules. On the prime-theoretic side, we relate strongly multiplicative sets to strongly prime ideals and strongly zero-dimensional rings, answer the Hamed--Malek question on the role of strong multiplicativity for chains of $S$-prime ideals, prove a strong Krull separation lemma together with a maximal-ideal correspondence for $R_S$, and connect the theory with the regular $m$-complement operator. In particular, every nontrivial strongly multiplicative localization must invert a zero divisor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16741",
  "author_total": 2,
  "authors": [
    {
      "name": "Hwankoo Kim"
    },
    {
      "name": "Suat Koç"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9db571gfr0cht.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16741",
  "fulltext_url": null,
  "id": "9db571gfr0cht",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16741",
  "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",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7213431512",
  "pdf_url": "https://arxiv.org/pdf/2609.16741",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9db571gfr0cht.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16741",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16741"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Strongly multiplicative sets, idempotent localizations, and $S$-prime phenomena",
  "title_text": "Strongly multiplicative sets, idempotent localizations, and S-prime phenomena",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/9db571gfr0cht",
  "venue": "arXiv",
  "versions": []
}