{
  "aliases": [
    "cotorsion",
    "cotorsion-module"
  ],
  "backlinks": null,
  "body": "\n## Summary\n\nIn [[abelian|071DDFVG4J]] group theory, an abelian group is said to be **cotorsion** if any extension of it by a torsion-free abelian group splits. If the group is $M$, this says that the Ext-group $\\operatorname{Ext}_\\mathbb{Z}(F,M)$ is zero for all torsion-free groups $F$. Since any such $F$ embeds into a direct sum of copies of $\\mathbb{Q}$, it suffices to check the condition on the group of [[rational numbers|0NCJ5WXN6Z]].\n",
  "commit": "a799a9852bcfeae9dc83faa3bef60208bb53686b",
  "front": {
    "aliases": [
      "cotorsion",
      "cotorsion-module"
    ],
    "inventory": [
      "Q5BHSM8Y72"
    ],
    "kind": "concept",
    "name": "Cotorsion group",
    "same_as": {
      "wikidata": "Q5175462",
      "wikipedia": "Cotorsion group"
    },
    "sources": [
      {
        "locator": "Cotorsion group",
        "mention": "wikipedia://en/5739241",
        "note": "adopted",
        "sections": [
          "Summary"
        ],
        "wikipedia": "en:5739241@r1366628056"
      }
    ],
    "tag": "CAAPB5WJSV"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/CAAPB5WJSV/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/CAAPB5WJSV.md",
  "msc": null,
  "name": "Cotorsion group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CAA/CAAPB5WJSV.md",
  "status": null,
  "tag": "CAAPB5WJSV",
  "uri": "m2://CAAPB5WJSV",
  "url": "https://math2.tamnd.com/wiki/cotorsion-group/",
  "writing": {
    "tag": "CAAPB5WJSV",
    "kind": "concept",
    "level": "W1",
    "next": [
      {
        "what": "edge defines in",
        "note": "at least one definition",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of in",
        "note": "at least two examples",
        "have": 0,
        "want": 2
      },
      {
        "what": "edge not_instance_of in",
        "note": "at least one non-example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge specializes both",
        "note": "the more general notions and the special cases",
        "have": 0,
        "want": 1
      }
    ]
  }
}