{
  "abstract": "Let $k$ be a field of characteristic zero, let $R$ be a commutative $k$-algebra, let $I$ be a proper ideal of $R$, and assume that $A=R/I$ is Noetherian. We prove that if $a=[f]$ in $A$ is nilpotent and $d_{A/k}a=0$, then $I:f$ is not contained in $(I,f)$. If $R$ is local with maximal ideal $\\mathfrak n$, then $I:f$ is not contained in $(I,f)+\\mathfrak n(I:f)$. As applications, we prove the isolated hypersurface case in arbitrary dimension and extend the conclusion to non-isolated critical loci on smooth varieties. Finally, we exhibit a five-variable isolated singularity satisfying $J_f:f\\subseteq\\overline{J_f}$, thereby disproving the integral-closure strengthening.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29516",
  "author_total": 1,
  "authors": [
    {
      "name": "Yizhi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ef8f2n6v9dec7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29516",
  "fulltext_url": null,
  "id": "ef8f2n6v9dec7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29516",
  "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"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7214372886",
  "pdf_url": "https://arxiv.org/pdf/2609.29516",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ef8f2n6v9dec7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29516",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29516"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Jordan Blocks, Differential Traces, and Colon-Ideal Noncontainment over Noetherian Quotients",
  "title_text": "Jordan Blocks, Differential Traces, and Colon-Ideal Noncontainment over Noetherian Quotients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/ef8f2n6v9dec7",
  "venue": "arXiv",
  "versions": []
}