{
  "abstract": "In analogy with the notion of Krull dimension, for every commutative ring $R$, we introduce the notion of pure dimension $\\pdim(R)$ by considering chains of purely-prime ideals. Then we see that the inequality $\\pdim(R)\\leqslant\\dim(R)$ holds for a very broad class of rings $R$, including pseudo-Noetherian rings (e.g., Noetherian rings), Gelfand rings (e.g., local rings), mp-rings (e.g., integral domains), and rings with finitely many maximal ideals or finitely many minimal primes. In fact, for all of these rings $R$, we always have $\\pdim(R)=0$, i.e., every purely-prime ideal of $R$ is purely-maximal. \\\\ However, we prove a classification theorem which states for each ordinal number $Ω$ the proper pure ideals of the commutative ring $$R_Ω=k[X_α : α<Ω]/\\big(X_α(1-X_β) : α<β<Ω\\big)$$ are exactly of the form $I_λ=(x_α: α<λ)$ with $λ=0$ or $λ\\leqslantΩ$ a limit ordinal (we write $x_α$ for the image of $X_α$ in $R_Ω$). In addition, all of them are purely-prime ideals and form the following unique strictly ascending chain: $$I_{0}=0\\subset I_ω\\subset I_{ω\\cdot2}\\subset I_{ω\\cdot 3}\\subset\\ldots\\subset I_{ω\\cdotα}$$ where $α$ is a unique ordinal with $Ω=ω\\cdotα+n$ and $n\\inω$. To prove this theorem, we use the full strength of ...",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14251",
  "author_total": 1,
  "authors": [
    {
      "name": "Abolfazl Tarizadeh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/26sdxj3qkby7d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14251",
  "fulltext_url": null,
  "id": "26sdxj3qkby7d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.14251",
  "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"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7203654670",
  "pdf_url": "https://arxiv.org/pdf/2608.14251",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/26sdxj3qkby7d.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2608.14251",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14251"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Pure dimension and counterexamples to the maximality of purely-primes",
  "title_text": "Pure dimension and counterexamples to the maximality of purely-primes",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/26sdxj3qkby7d",
  "venue": "arXiv",
  "versions": []
}