{
  "abstract": "Understanding the relation between $\\dim R$ and $\\dim R[[x]]$ is a classical problem in commutative algebra. For a Noetherian ring $R$, one has $\\dim R[[x]]=\\dim R+1$, but the general case is considerably more delicate. In 1973, Arnold proved that finite power-series dimension requires the strong finite type (SFT) condition, whereas, in 2002, Coykendall constructed a one-dimensional SFT domain whose power series ring has infinite dimension. The question of Coykendall and Gilmer whether $\\dim R[[x]]<\\infty$ forces $\\dim R[[x]]\\le2\\dim R+1$ was answered negatively by Kang and Park in 2009. In this paper, we prove that, as $R$ ranges over the nonzero commutative rings with identity, the finite pairs $(\\dim R,\\dim R[[x]])$ are exactly $(0,1)$ and the pairs $(n,m)$ with $1\\le n<m$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17854",
  "author_total": 3,
  "authors": [
    {
      "name": "Viet-Hoang Tran"
    },
    {
      "name": "Thieu N. Vo"
    },
    {
      "name": "Tan M. Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/82kk8kzta6tst.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17854",
  "fulltext_url": null,
  "id": "82kk8kzta6tst",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17854",
  "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"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7213529283",
  "pdf_url": "https://arxiv.org/pdf/2609.17854",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/82kk8kzta6tst.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17854",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17854"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Dimensions of a ring and its formal power series ring",
  "title_text": "Dimensions of a ring and its formal power series ring",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/82kk8kzta6tst",
  "venue": "arXiv",
  "versions": []
}