{
  "abstract": "Let $R$ be a finite associative ring with identity, and let $\\Nul(R)$ denote the set of polynomials in a central indeterminate that vanish identically on $R$ under right evaluation. We establish two results concerning the failure of $\\Nul(R)$ to be a right ideal. First, if $e$ is an idempotent, $f=1-e$, and either $eRf=0$ or $fRe=0$, then right multiplication by $e$ preserves $\\Nul(R)$. Thus both off-diagonal Peirce components must be nonzero whenever $e$ witnesses a failure of right stability. Second, if $J$ is a finite nilpotent ideal whose additive group is a $2$-group and $e,f$ are complementary idempotents such that \\[ eJf\\neq0,\\qquad fJe\\neq0,\\qquad J^3\\neq0, \\] then $|J|\\geq32$. Werner's theorem that $\\Jac(R)^3=0$ implies the two-sidedness of $\\Nul(R)$ therefore yields $|R|\\geq128$ whenever $\\Nul(R)$ is not two-sided. We also construct a tiled matrix ring of characteristic $4$ and order $128$ whose null ideal is not two-sided.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01150",
  "author_total": 1,
  "authors": [
    {
      "name": "Hongfeng Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/523c8kpb9cnkf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01150",
  "fulltext_url": null,
  "id": "523c8kpb9cnkf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01150",
  "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.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7206178840",
  "pdf_url": "https://arxiv.org/pdf/2609.01150",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/523c8kpb9cnkf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01150",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01150"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Peirce Stability of Null Polynomials and a Sharp Radical Bound for Finite Rings",
  "title_text": "Peirce Stability of Null Polynomials and a Sharp Radical Bound for Finite Rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/523c8kpb9cnkf",
  "venue": "arXiv",
  "versions": []
}