{
  "abstract": "Let \\(H\\) be a multiplicative group, \\(R\\) a commutative \\(H\\)-graded ring, and \\(M\\) a nonzero finitely generated gr-semisimple \\(H\\)-graded \\(R\\)-module. We assume that \\(M\\) admits a fixed decomposition into powers of gr-simple modules with pairwise distinct annihilators, each repeated block carrying the componentwise grading. For a nonzero homogeneous element \\(x\\), we compute the colon ideal \\((Rx:M)\\) and show that it is determined by the multiplicity-one blocks on which \\(x\\) has a nonzero component. This leads to the criterion \\( x\\leftrightarrow y \\text{ if and only if } σ(x)\\capσ(y)=\\varnothing, \\) where \\(x\\leftrightarrow y\\) means that \\(x\\) and \\(y\\) are adjacent in \\(Γ_H(M)\\). Thus adjacency is governed by disjointness of multiplicity-one obstruction supports. The corresponding obstruction classes form a weighted disjointness structure: higher multiplicities affect the sizes of the classes, but not the adjacency relation between their supports. We use this description to study graph invariants, reconstruction, and Kneser-type layers, and, in the finite case, to obtain formulas for the obstruction-class sizes and a reduction of the adjacency spectrum.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14003",
  "author_total": 3,
  "authors": [
    {
      "name": "M. Aijaz"
    },
    {
      "name": "Shahnawaz A. Rather"
    },
    {
      "name": "S Pirzada"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f00ceh9f5pebx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14003",
  "fulltext_url": null,
  "id": "f00ceh9f5pebx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14003",
  "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": "W7221825520",
  "pdf_url": "https://arxiv.org/pdf/2610.14003",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f00ceh9f5pebx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14003",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14003"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Multiplicity-One Structure in Graded Zero-Divisor Graphs",
  "title_text": "Multiplicity-One Structure in Graded Zero-Divisor Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/f00ceh9f5pebx",
  "venue": "arXiv",
  "versions": []
}