{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1Z9MRNNYYD"
    },
    {
      "field": "uses",
      "target": "7938EKMMVR"
    },
    {
      "field": "uses",
      "target": "7MPCVNQ8SF"
    },
    {
      "field": "uses",
      "target": "AA943HVK6Z"
    },
    {
      "field": "uses",
      "target": "C7Q2MVD933"
    },
    {
      "field": "defines",
      "target": "90ASENKFMH"
    }
  ],
  "body": "\n## Summary\n\nA fractional ideal of an integral domain $R$ is a subset of its field of fractions.\nAfter clearing denominators, it becomes an [[ideal|096Z9MP2ME]] of $R$.\nFractional ideals can be multiplied, and in a [[Dedekind domain|7NPY2CN5KB]] every nonzero fractional ideal has an inverse.\nSo the nonzero fractional ideals of a Dedekind domain form a group, and its quotient by the principal ones is the [[ideal class group|WQ46BD3ARY]].\n\n## Explanations\n\n### L1 human\n\nIdeals of the integers are the sets $n\\Z$ of multiples of a number $n$.\nThey can be multiplied, $(2\\Z)(3\\Z) = 6\\Z$, but not divided: there is no ideal $I$ of $\\Z$ with $(2\\Z) I = \\Z$.\nFractional ideals allow fractions: $\\tfrac12 \\Z = \\set{\\ldots, -\\tfrac12, 0, \\tfrac12, 1, \\ldots}$ is one, and $(2\\Z)(\\tfrac12\\Z) = \\Z$.\nSo fractional ideals are to ideals what fractions are to integers.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13A15",
      "11R04"
    ],
    "name": "Fractional ideal",
    "same_as": {
      "wikidata": "Q1497184",
      "wikipedia": "Fractional ideal"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fractional_ideal"
      }
    ],
    "tag": "YB4JTB4WJN"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/YB4JTB4WJN/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/YB4JTB4WJN.md",
  "msc": [
    "13A15",
    "11R04"
  ],
  "name": "Fractional ideal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YB4/YB4JTB4WJN.md",
  "status": null,
  "tag": "YB4JTB4WJN",
  "uri": "m2://YB4JTB4WJN",
  "url": "https://math2.tamnd.com/wiki/fractional-ideal/",
  "writing": {
    "tag": "YB4JTB4WJN",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}