{
  "abstract": "Let $R$ be a $ϕ$-ring and put $N=Nil(R)$. We determine the $ϕ$-weak global dimension of every $ϕ$-Prüfer ring. Its only possible values are $0$, $1$, and $\\infty$: the value is $0$ when $R/N$ is a field, it is $1$ when $R$ is strong and $R/N$ is a nonfield Prüfer domain, and it is infinite when $R$ is not strong. No nilpotence assumption on $N$ is needed. The proof is based on the canonical kernel $K=ker(R\\toϕ(R))$. In the local non-strong case, $K$ is a nonzero torsion divisible module over the valuation domain $R/N$, and its self-Tor groups satisfy \\[ Tor^R_{2n}(K,K)=0\\quad(n\\geq 1), \\qquad Tor^R_{2n+1}(K,K)\\neq 0\\quad(n\\geq 0). \\] It follows that $fd_RK=\\infty$, and a filtered-colimit argument transfers this obstruction to cyclic quotients by nonnil principal ideals. We also prove the parallel trichotomy for the $ϕ$-global dimension of $ϕ$-Dedekind rings. For strong $ϕ$-rings we obtain a change-of-rings theorem without assuming that the quotient map $R\\to R/N$ splits. If $D=R/N$, $Q=Frac(D)$ and $T=T(R)$ is the total quotient ring, then \\[ R\\cong T\\times_Q D. \\] Using Ferrand's flat patching theorem and Milnor's projective patching theorem, we prove that every torsion $D$-module $M$, regarded as an $R$-module through $R\\to D$, satisfies \\[ fd_R(M)=fd_D(M), \\qquad pd_R(M)=pd_D(M). \\] Consequently, for every strong $ϕ$-ring, \\[ϕ-wgd(R)=wgld(R/N), \\qquad ϕ-gld(R)=gld(R/N). \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05567",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaolei Zhang"
    },
    {
      "name": "Wei Qi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b7vvbcg58wh29.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05567",
  "fulltext_url": null,
  "id": "b7vvbcg58wh29",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05567",
  "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"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7220691912",
  "pdf_url": "https://arxiv.org/pdf/2610.05567",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b7vvbcg58wh29.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05567",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05567"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the $ϕ$-Weak Global Dimensions and $ϕ$-Global Dimensions",
  "title_text": "On the ϕ-Weak Global Dimensions and ϕ-Global Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/b7vvbcg58wh29",
  "venue": "arXiv",
  "versions": []
}