{
  "abstract": "Let $R$ be a commutative ring. A submodule $N$ of an $R$-module $M$ is semiprime if $m\\in(N:m)M$ implies $m\\in N$. It is known that every semiprime submodule of $R^n$ is an intersection of prime submodules, and Aryapoor extended this to finitely generated modules. We show that no finiteness assumption is needed: every semiprime submodule of an \\emph{arbitrary} $R$-module is the intersection of the prime submodules containing it. The proof is short and uses neither coordinates nor presentations; its key step is that a nonzero element $q$ of a semiprime module $Q$ satisfies $sq\\notin\\p Q$ for all $s\\notin\\p$, where $\\p$ is any minimal prime over the annihilator of $q$. We define an $R$-module $M$ to be real if $\\sum_i m_i\\otimes m_i=0$ in $M\\otimes_RM$ implies $m_i=0$ for all $i$, and a submodule $N\\subseteq M$ real if the module $M/N$ is real. We prove the corresponding result for real submodules: every real submodule of an arbitrary module is the intersection of the real prime submodules containing it. On $R^n$ our notion of realness coincides with the coordinate definition used in earlier work on real Nullstellensätze for free modules.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15869",
  "author_total": 1,
  "authors": [
    {
      "name": "Jakob Cimprič"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ckb3d8cpcfadp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15869",
  "fulltext_url": null,
  "id": "ckb3d8cpcfadp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15869",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7221754783",
  "pdf_url": "https://arxiv.org/pdf/2610.15869",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ckb3d8cpcfadp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15869",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15869"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Semiprime and real submodules of arbitrary modules are intersections of prime and real prime submodules",
  "title_text": "Semiprime and real submodules of arbitrary modules are intersections of prime and real prime submodules",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/ckb3d8cpcfadp",
  "venue": "arXiv",
  "versions": []
}