{
  "abstract": "In this paper, we study regularity and clean-type decompositions for matrices over dual numbers. We characterize the von Neumann regularity of dual matrices by establishing a necessary and sufficient compatibility condition between their real and dual components. At the element level, we determine the conditions under which an individual dual matrix $\\mathcal{M} = A + Bε$ is regular, $π$-regular, nil-clean, or strongly nil-clean. Using the solvability of Sylvester matrix equations, we show that the matrix ring over dual numbers $M_n(\\mathbb{D})$ is strongly $π$-regular and strongly clean. Moreover, $M_n(\\mathbb{D})$ is $π$-regular, clean, $r$-clean, strongly $r$-clean, $\\mathrm{NR}$-clean, and strongly $\\mathrm{NR}$-clean. On the other hand, $M_n(\\mathbb{D})$ fails to be regular, strongly regular, nil-clean, strongly nil-clean, and uniquely $\\mathrm{NR}$-clean, although it inherits several strong decomposition properties from $M_n(\\mathbb{R})$. Our characterizations at the element level show exactly when each property holds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29468",
  "author_total": 1,
  "authors": [
    {
      "name": "Berrin Şentürk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8v7gpf8d39rxf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29468",
  "fulltext_url": null,
  "id": "8v7gpf8d39rxf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29468",
  "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": "W7214295436",
  "pdf_url": "https://arxiv.org/pdf/2609.29468",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8v7gpf8d39rxf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.29468",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29468"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Regularity and Clean Properties of Matrices over Dual Numbers",
  "title_text": "On the Regularity and Clean Properties of Matrices over Dual Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/8v7gpf8d39rxf",
  "venue": "arXiv",
  "versions": []
}