{
  "abstract": "Let $A$ be an abelian group and $E=\\Endo_{\\Z}(A)$. We give a module-theoretic description of the singular $E$-submodules asked for in Fuchs' Problem~1.2. For a unital ring $R$ and a left $R$-module $M$, put $W=I_R({}_RR)\\oplus I_R(M)$ and $J=\\Jaco(\\Endo_R(W))$, and let $u$ be the image of $1_R$. The classical essential-kernel criterion yields \\[ Z_R(M)=M\\cap Ju. \\] For $R=E$ and $M=A$, all singular submodules are therefore the $E$-submodules of $A\\cap Ju$. Writing $T=t(A)$ and $B=A/T$, we prove a torsion-transfer formula, identify the torsion part as $\\bigoplus_p pT_p$, and describe simultaneous prime lifting by a canonical obstruction. The resulting extension gives an $\\Extt/\\Homm$ parametrization of all singular submodules. We obtain explicit formulas for torsion groups and for $\\Z(p^\\infty)\\oplus B$ with $B$ torsion-free; in the latter case the fully invariant subgroup lattice of $B$ occurs as an interval. The general description retains the induced endomorphism action and extension data, rather than giving a classification by classical group invariants.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21832",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiang Yang"
    },
    {
      "name": "Xin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7w19446n9hkp5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21832",
  "fulltext_url": null,
  "id": "7w19446n9hkp5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21832",
  "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.GN"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7213927067",
  "pdf_url": "https://arxiv.org/pdf/2609.21832",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7w19446n9hkp5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21832",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21832"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Singular Submodules of Abelian Groups over Their Endomorphism Rings",
  "title_text": "Singular Submodules of Abelian Groups over Their Endomorphism Rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/7w19446n9hkp5",
  "venue": "arXiv",
  "versions": []
}