{
  "abstract": "For a commutative ring $R$, we call an $R$-module $B$ Baer if $\\Ext_R^1(B,T)=0$ for torsion $R$-module $T$. It is known that every Baer module is projective when $R$ is $τ_q$-semisimple, i.e., rings whose total rings of quotients are semisimple. And it was conjectured that the converse characterizes $τ_q$-semisimple rings. We show that the converse fails in two substantially different ways. First, the failure is already systematic in the non-reduced Noetherian case. If $D$ is a Dedekind domain which is not a field and $R=D[\\varepsilon]/(\\varepsilon^2)$, then every Baer $R$-module is projective, although $R$ is not reduced, and thus is not $τ_q$-semisimple. Second, the converse fails even among reduced rings, and in fact among reduced coherent Bézout rings. The main tool is a countable-ring criterion. For a countable Noetherian domain $D$, let $\\EC(D)$ be the ring of eventually constant sequences over $D$. If $D$ is not a field, then every Baer $\\EC(D)$-module is projective, whereas $\\EC(D)$ is reduced and has countably infinitely many minimal prime ideals, and thus is not $τ_q$-semisimple.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14338",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaolei Zhang"
    },
    {
      "name": "Guocheng Dai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cgbtwxg5yw8nb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14338",
  "fulltext_url": null,
  "id": "cgbtwxg5yw8nb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14338",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.14338",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cgbtwxg5yw8nb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14338",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14338"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Baer Splitting Beyond $τ_q$-Semisimplicity",
  "title_text": "Baer Splitting Beyond τ_q-Semisimplicity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/cgbtwxg5yw8nb",
  "venue": "arXiv",
  "versions": []
}