{
  "abstract": "We study the additive and multiplicative arithmetic of algebraic monogenic semidomains of the form $S_α$. We first relate their structure to the conjugates of $α$, characterize when $S_α$ is a ring, and determine when its additive monoid is free. We then investigate units and prove a Dirichlet-type theorem showing that $S_α^\\times$ is the product of a finite cyclic group of roots of unity and a finitely generated free abelian group; in particular, the unit group is free abelian when $α$ is positive. We also give a criterion that produces non-atomic algebraic monogenic semidomains. Turning to finiteness properties, we prove that $\\mathbb{Z}[α]$ is a finite factorization domain for every algebraic number $α$, establish criteria ensuring that $S_α$ is a finite factorization semidomain, and construct infinitely many positive cubic generators yielding finite factorization semidomains that are not rings. Finally, we give a membership criterion for rational elements of $\\mathbb{Z}[β]$ for arbitrary algebraic $β$, and prove that, for algebraic integers $α$, unique factorization ascends from $\\mathbb{Z}[α]$ to $\\mathbb{Z}[α/n]$ for every $n\\in\\mathbb{N}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32138",
  "author_total": 4,
  "authors": [
    {
      "name": "Jiya Dani"
    },
    {
      "name": "Felix Gotti"
    },
    {
      "name": "Bryan Li"
    },
    {
      "name": "Arav Paladiya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5h6n23z1styw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32138",
  "fulltext_url": null,
  "id": "c5h6n23z1styw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32138",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7214761413",
  "pdf_url": "https://arxiv.org/pdf/2609.32138",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5h6n23z1styw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32138",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32138"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Factorizations in algebraic monogenic semidomains",
  "title_text": "Factorizations in algebraic monogenic semidomains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/c5h6n23z1styw",
  "venue": "arXiv",
  "versions": []
}