{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $R$ be a [[Dedekind domain|7NPY2CN5KB]] that is also a [[unique factorization domain|0W4X9G6263]].\nThen $R$ is a [[principal ideal domain|0CMD0BCH8T]].\n\n## Remarks\n\nLet $P$ be a nonzero [[prime ideal|00MHHZ9SNE]] and let $a \\in P$ be nonzero.\nThe element $a$ is not a unit, and one of its [[irreducible|0YMT0D2442]] factors $\\pi$ lies in $P$.\nIn a UFD, $\\pi$ is a [[prime element|01VH21EV6F]], so $(\\pi)$ is a nonzero prime ideal inside $P$; as nonzero primes of a Dedekind domain are maximal, $(\\pi) = P$.\nSo every nonzero prime is principal, and by the [[unique factorization of ideals|S6PHT75MEK]] every nonzero ideal is a product of principal ideals, hence principal.\n\nExample: $\\Z[\\sqrt{-5}]$ is a Dedekind domain, but $6 = 2 \\cdot 3 = (1 + \\sqrt{-5})(1 - \\sqrt{-5})$ gives two different factorizations into irreducibles, so it is not a UFD; consistent with this, $(2, 1 + \\sqrt{-5})$ is not principal.\nThe Dedekind hypothesis cannot be dropped: $k[x, y]$ for a field $k$ is a UFD, but the ideal $(x, y)$ is not principal.\n\n## Explanations\n\n### L1 human\n\nFor rings of algebraic integers, unique factorization of numbers and \"every ideal has one generator\" are the same property.\nOne direction holds in every ring: a principal ideal domain always has unique factorization.\nThis result gives the other direction for Dedekind domains.\nSo a ring of integers of a number field has unique factorization exactly when its ideals are all principal.\n\n### L2 human\n\nIn terms of the [[ideal class group|WQ46BD3ARY]], a Dedekind domain is a UFD exactly when its class group is trivial, and this is the same as being a PID.\nThe class group thus measures the failure of unique factorization of elements.\nFor example, the class group of $\\Z[\\sqrt{-5}]$ has order $2$.\nThe proof uses only that nonzero primes are maximal and that ideals factor into primes, so it is a good first exercise with the ideal factorization theorem.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "7NPY2CN5KB",
      "0W4X9G6263",
      "0CMD0BCH8T"
    ],
    "kind": "statement",
    "msc": [
      "13F05",
      "13F15",
      "13F10"
    ],
    "name": "A Dedekind UFD is a PID",
    "role": "proposition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dedekind_domain"
      }
    ],
    "tag": "YGV5S3MRD6",
    "uses": [
      "7NPY2CN5KB",
      "0W4X9G6263",
      "0CMD0BCH8T",
      "00MHHZ9SNE",
      "0YMT0D2442",
      "01VH21EV6F",
      "S6PHT75MEK"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/YGV5S3MRD6/",
  "links": [
    {
      "field": "about",
      "target": "7NPY2CN5KB"
    },
    {
      "field": "about",
      "target": "0W4X9G6263"
    },
    {
      "field": "about",
      "target": "0CMD0BCH8T"
    },
    {
      "field": "uses",
      "target": "7NPY2CN5KB"
    },
    {
      "field": "uses",
      "target": "0W4X9G6263"
    },
    {
      "field": "uses",
      "target": "0CMD0BCH8T"
    },
    {
      "field": "uses",
      "target": "00MHHZ9SNE"
    },
    {
      "field": "uses",
      "target": "0YMT0D2442"
    },
    {
      "field": "uses",
      "target": "01VH21EV6F"
    },
    {
      "field": "uses",
      "target": "S6PHT75MEK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YGV5S3MRD6.md",
  "msc": [
    "13F05",
    "13F15",
    "13F10"
  ],
  "name": "A Dedekind UFD is a PID",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YGV/YGV5S3MRD6.md",
  "status": null,
  "tag": "YGV5S3MRD6",
  "uri": "m2://YGV5S3MRD6",
  "url": "https://math2.tamnd.com/wiki/a-dedekind-ufd-is-a-pid/",
  "writing": {
    "tag": "YGV5S3MRD6",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}