{
  "abstract": "Let $K$ be any field and $E$ any directed graph. We characterize up to isomorphism the simple (i.e., minimal) left ideals of the Leavitt path algebra $L_K(E)$. Then, for each simple %(i.e., minimal) left ideal $I$ of %the Leavitt path algebra $L_K(E)$, we explicitly construct the injective envelope of $I$. This result generalizes to all graphs $E$ and all simple left ideals in $L_K(E)$ the construction presented previously by the three authors for the specific case of the Jacobson algebra $R=K\\langle X,Y | XY=1\\rangle$ and the simple left $R$-ideal $R(1-YX)$. Our method involves defining an $L_K(E)$-module structure on a $K$-vector space of infinite series. We conclude the article by showing how our construction directly gives a description of the injective envelope of simple $L_K(E)$-modules arising from two types of infinite emitters in $E$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00947",
  "author_total": 3,
  "authors": [
    {
      "name": "G. Abrams"
    },
    {
      "name": "F. Mantese"
    },
    {
      "name": "A. Tonolo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bsrd688bx217g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00947",
  "fulltext_url": null,
  "id": "bsrd688bx217g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00947",
  "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": "W7206203561",
  "pdf_url": "https://arxiv.org/pdf/2609.00947",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bsrd688bx217g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.00947",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00947"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The injective envelope of simple modules over Leavitt path algebras I: simple left ideals",
  "title_text": "The injective envelope of simple modules over Leavitt path algebras I: simple left ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/bsrd688bx217g",
  "venue": "arXiv",
  "versions": []
}