{
  "id": "2404.15769",
  "version": "v2",
  "published": "2024-04-24T09:44:18.000Z",
  "updated": "2024-05-27T13:15:37.000Z",
  "title": "Realization of monoids with countable sum",
  "authors": [
    "Zahra Nazemian"
  ],
  "comment": "25 Pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "For every infinite cardinal number $\\kappa$, $\\kappa$-monoids and their realization have recently been introduced and studied by Nazemian and Smertnig. A $\\kappa$-monoid $H$ has a realization to a ring $R$ if there exists an element $x \\in H$ such that $H$ is $\\aleph_1 ^{-}$-braided over $\\text{add}(\\aleph_0 x)$, and $\\text{add}(\\aleph_0 x)$, as $\\aleph_0$-monoid, has a realization to $R$. Furthermore, $H$ has a realization to hereditary rings if there exists an element $x \\in H$ such that $H$ is braided over $\\text{add}(x)$. These prompt an investigation into when $\\aleph_0$-monoids have realizations. In this paper, we discuss the realization of $\\aleph_0$-monoids and provide a complete characterization for the realization of two-generated ones in hereditary Von Neumann regular rings.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-05-27T13:15:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16D70",
      "16D40",
      "16E50",
      "16P10",
      "16S99"
    ],
    "keywords": [
      "realization",
      "countable sum",
      "hereditary von neumann regular rings",
      "infinite cardinal number",
      "hereditary rings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}