{
  "abstract": "This paper investigates reversed product properties of two elements in rings. Motivated by Cline's formula, we characterize reversible and $\\ast$-reversible rings in terms of group invertible elements, EP elements, and the transfer behaviour of generalized inverses for reversed products. We prove that a unital ring $R$ is reversible if and only if $ab\\in R^{\\sharp}$ yields $ba\\in R^{\\sharp}$. For an involutive ring $R$, $R$ is $\\ast$-reversible precisely whenever $ab\\in R^{\\mathrm{EP}}$ implies $b^{\\ast}a\\in R^{\\mathrm{EP}}$. Several counterexamples are constructed to differentiate these ring classes, and the mutual inclusion relations among them are also discussed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12643",
  "author_total": 3,
  "authors": [
    {
      "name": "Long Wang"
    },
    {
      "name": "Tingting Li"
    },
    {
      "name": "Yuheng Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cx19ksf692qhh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12643",
  "fulltext_url": null,
  "id": "cx19ksf692qhh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12643",
  "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": "W7212539257",
  "pdf_url": "https://arxiv.org/pdf/2609.12643",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cx19ksf692qhh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12643",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12643"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Notes on Reversed Products of Two Elements in Rings",
  "title_text": "Notes on Reversed Products of Two Elements in Rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/cx19ksf692qhh",
  "venue": "arXiv",
  "versions": []
}