{
  "abstract": "Let $R(A,X)$ denote the largest ideal of a primitive axial algebra $(A,X)$ that contains no axis from the specified generating set $X$, and let $J(A)$ be the intersection of the maximal ideals of $A$. Mamontov, Shpectorov, and Zhelyabin asked whether $R(A,X)=J(A)$ always holds. We give a negative answer. Over every field of characteristic different from $2$, the two-dimensional commutative algebra with basis $a,b$ and multiplication $a^2=a$, $ab=2b$, and $b^2=b$ is a primitive axial algebra for an explicit fusion law and the generating set $X={a,b}$. Its complete ideal lattice is $0<\\mathbb{F}b<A$, whence $R(A,X)=0$ and $J(A)=\\mathbb{F}b$; in particular, the primitive axis $b$ lies in $J(A)$. Over $\\mathbb{C}$, this axial presentation is equivalent to the previously classified $D(-1),{e_2,a_6}$ presentation with fusion law $F_{D3}$. Thus the algebra and axial structure are known; the new point is the computation of its Jacobson radical and the resulting counterexample to the radical-equality question.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28653",
  "author_total": 1,
  "authors": [
    {
      "name": "Bo Peng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1y035kmxkjc2r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28653",
  "fulltext_url": null,
  "id": "1y035kmxkjc2r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28653",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2608.28653",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1y035kmxkjc2r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.28653",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28653"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Two-Dimensional Counterexample to Radical Equality in Primitive Axial Algebras",
  "title_text": "A Two-Dimensional Counterexample to Radical Equality in Primitive Axial Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/1y035kmxkjc2r",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.5281/zenodo.22020622",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}