{
  "abstract": "We study direct products of monoids with zero and give a criterion under which their factors can be recovered from the multiplicative structure alone. While classical decomposition theory encodes direct products through factor congruences, central elements, and refinement properties, we give a concrete multiplicative reconstruction mechanism. If the factors have no nontrivial complemented central idempotents, then the coordinate idempotents are precisely the atoms and coatoms of the complemented-central-idempotent poset, and their multiplicative stabilizers are precisely the coordinate factors and cofactors. It follows that every isomorphism between such products is monomial, yielding the corresponding wreath-product description of automorphism groups. More generally, every product decomposition is obtained by grouping the original factors; in particular, strict refinement follows. We apply these results to multiplicative monoids of directly indecomposable unital rings, including connected commutative rings, and to arithmetic examples arising from residue-class rings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08815",
  "author_total": 3,
  "authors": [
    {
      "name": "Joseph Atalaye"
    },
    {
      "name": "Liam Baker"
    },
    {
      "name": "Sophie Marques"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dxn7wwd1b2s5n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08815",
  "fulltext_url": null,
  "id": "dxn7wwd1b2s5n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08815",
  "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.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7211997823",
  "pdf_url": "https://arxiv.org/pdf/2609.08815",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dxn7wwd1b2s5n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.08815",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08815"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Direct-product rigidity and factor reconstruction for monoids with zero",
  "title_text": "Direct-product rigidity and factor reconstruction for monoids with zero",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/dxn7wwd1b2s5n",
  "venue": "arXiv",
  "versions": []
}