{
  "abstract": "The aim of the paper is to study Euclidean and l-finite Euclidean modules over a ring, submodules of their finite direct sums, reduced and strong reduced echelon forms of matrices with coefficients in Euclidean modules and bimodules. The concept of Euclidean module is a module-theoretic analogue of the concept of Euclidean ring. The class of (not necessarily commutative) Euclidean rings is a nice but `very small' class of rings that have arithmetic-like properties. On the contrary, surprisingly, the class of Euclidean modules is a `large class' of modules. The Euclidean modules exist for many {\\em constructions} of rings that are defined over {\\em arbitrary} rings. For example, for every skew polynomial ring $A=D[x;\\s, \\d]$ with coefficients in an {\\em arbitrary} ring $D$, the generalized Weyl algebra $D[x,y;\\s, a]$, the Weyl algebra $A_1=K[x][\\der; \\frac{d}{dx}]$, $U(\\sl2)$ and many classical algebras, there exist Euclidean modules.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37893",
  "author_total": 1,
  "authors": [
    {
      "name": "V. V. Bavula"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/613j93bdv9c46.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37893",
  "fulltext_url": null,
  "id": "613j93bdv9c46",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37893",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7214891858",
  "pdf_url": "https://arxiv.org/pdf/2609.37893",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/613j93bdv9c46.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37893",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37893"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Euclidean modules, submodules of their direct sums and strong reduced echelon form",
  "title_text": "Euclidean modules, submodules of their direct sums and strong reduced echelon form",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/613j93bdv9c46",
  "venue": "arXiv",
  "versions": []
}