{
  "abstract": "In 1978, Amitsur and Small asked whether, for every division ring $D$, maximal left ideals of $D[x_1,\\ldots,x_n]$ contract to maximal left ideals in smaller polynomial subrings. Chapman and the author showed that the answer is negative in general and called $D$ an Amitsur-Small ring when this contraction property always holds. Earlier work showed that Hamilton's real quaternion algebra is Amitsur-Small, that division algebras of degree three are not Amitsur-Small, and that degree-two examples are restricted to a specific quaternionic form; subsequent work excluded cyclic division algebras of odd prime degree. We give a complete resolution in the centrally finite case. If $D$ has finite dimension over its center $F$, then $D$ is an Amitsur-Small ring if and only if either $D=F$, or $F$ is real closed and $D$ is the Hamilton quaternion algebra $(-1,-1)_F$. In every other noncommutative centrally finite case, failure already occurs in two variables: there is a maximal left ideal $M\\subseteq D[x,y]$ such that $M\\cap D[x]$ is not maximal in $D[x]$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04317",
  "author_total": 1,
  "authors": [
    {
      "name": "Elad Paran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/11w8xvxfjwq1s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04317",
  "fulltext_url": null,
  "id": "11w8xvxfjwq1s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04317",
  "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.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7220660469",
  "pdf_url": "https://arxiv.org/pdf/2610.04317",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/11w8xvxfjwq1s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04317",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04317"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Characterization of the centrally finite Amitsur-Small division rings",
  "title_text": "Characterization of the centrally finite Amitsur-Small division rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/11w8xvxfjwq1s",
  "venue": "arXiv",
  "versions": []
}