{
  "abstract": "We define the integral profiles of all modules and introduce integral coefficient rings ${^{\\mathscr{P}}\\mkern-6.5mu\\text{\\&}\\mkern-5.5mu{_\\mathscr{I}}}(A)$ for all finite-dimensional complex algebras $A$. The integral profile of a module is a matrix with parameters. We provide a classification theorem for modules, to be precise, (1) two modules $M\\cong N$ are isomorphic if and only if their integral profiles are similar, i.e., $M\\cong N$ if and only if $\\displaystyle \\int M \\sim \\int N$. That is, the integral profile is a complete invariant of finite-dimensional modules. Furthermore, we show the following results in this paper: (2) we introduce the central integrals of algebras and show that it is isomorphic to the center of algebras; (3) we provide a descriptions for some special modules; (4) integral coefficient ring of $A$ (with a compatible orthogonal fixed embedding system) has Morita invariance; (5) the global dimension of $A$ is finite if and only if the embedded integral profile of $\\mathrm{top}(A)$ lies in ${^{\\mathscr{P}}\\mkern-6.5mu\\text{\\&}\\mkern-5.5mu{_\\mathscr{I}}}(A)[x]$; (6) the finitistic dimension of $A$ is finite if and only if each embedded integral profile of $M$ lying in ${^{\\mathscr{P}}\\mkern-6.5mu\\text{\\&}\\mkern-5.5mu{_\\mathscr{I}}}(A)[x]$ implies that its degree is less than or equal to a fixing integer $d\\in\\mathbb{N}^+$; (7) we provide two sufficient conditions, such that if a finite-dimensional complex algebra $A$ satisfies one of them, then its finitistic dimension is finite.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00143",
  "author_total": 1,
  "authors": [
    {
      "name": "Yu-Zhe Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vt3wbe9sagr4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00143",
  "fulltext_url": null,
  "id": "3vt3wbe9sagr4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00143",
  "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.CT"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7206171828",
  "pdf_url": "https://arxiv.org/pdf/2609.00143",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vt3wbe9sagr4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.00143",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00143"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Integral coefficient rings and homological dimensions of algebras",
  "title_text": "Integral coefficient rings and homological dimensions of algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/3vt3wbe9sagr4",
  "venue": "arXiv",
  "versions": []
}