{
  "abstract": "We continue the study of the basic theory of object-unital groupoid graded rings. In this work, we are especially interested in nilpotency conditions on the graded Jacobson radical. We introduce the concept of left/right objectwise nilpotency of graded ideals, and prove that this condition is appropriate for obtaining graded generalizations of the Hopkins--Levitzki theorem. Although this condition is not symmetric, we show that its two-sided version is suitable for defining gr-semiprimary rings. It is known that one-sided $Γ_0$-artinian rings need not be $Γ_0$-noetherian, but using our tools we prove that one-sided gr-hereditary $Γ_0$-artinian rings, two-sided $Γ_0$-artinian rings, and $d$-finitely generated one-sided $Γ_0$-artinian rings are $Γ_0$-noetherian. However, the first class need not be gr-semiprimary, whereas the other two always are. We illustrate our results with several (counter)examples, especially involving graded upper triangular matrices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00305",
  "author_total": 3,
  "authors": [
    {
      "name": "Zaqueu Cristiano"
    },
    {
      "name": "Wellington Marques de Souza"
    },
    {
      "name": "Javier Sánchez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3sx2zj1fpnmzj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00305",
  "fulltext_url": null,
  "id": "3sx2zj1fpnmzj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00305",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CT"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7206158052",
  "pdf_url": "https://arxiv.org/pdf/2609.00305",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3sx2zj1fpnmzj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00305",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00305"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Hopkins-Levitzki Type Theorems for Groupoid Graded Rings",
  "title_text": "Hopkins-Levitzki Type Theorems for Groupoid Graded Rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/3sx2zj1fpnmzj",
  "venue": "arXiv",
  "versions": []
}