{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1HD3EK5392"
    }
  ],
  "body": "\n## Statement\n\nLet $A$ be a [[Noetherian|0MVH4Y1CRZ]] [[integral domain|09S3PFBEPM]].\nLet $\\mathcal{P}$ be a set of [[prime elements|0MNHXBSX3M]] of $A$.\nLet $S$ be the [[multiplicative set|AN9SKEZ3VY]] generated by $\\mathcal{P}$: the products of a unit of $A$ and finitely many elements of $\\mathcal{P}$.\nIf the [[localization|4C1AE84N5N]] $S^{-1}A$ is a [[unique factorisation domain|06YMXJMXYS]], then $A$ is a unique factorisation domain.\n\n## Remarks\n\nSince $A$ is Noetherian, every nonzero nonunit of $A$ is a product of irreducible elements, so it suffices to show that every irreducible element $a$ of $A$ is prime.\nIf some prime $p \\in \\mathcal{P}$ divides $a$, then $a$ is associate to $p$ and so is prime.\nOtherwise one checks that $a$ is prime in $S^{-1}A$ and that $aS^{-1}A \\cap A = aA$, using that no prime of $\\mathcal{P}$ divides $a$.\nThen $aA$ is the contraction of a prime ideal, hence prime.\nExample: let $A = \\Z[x]$ and let $S$ be the set of nonzero integers.\n$S$ is generated by the primes $p$, which are prime in $\\Z[x]$ because $\\Z[x]/(p) \\cong \\F_p[x]$ is a domain.\nThen $S^{-1}A = \\Q[x]$ is a principal ideal domain, hence a UFD, so $\\Z[x]$ is a UFD.\nThe hypothesis on $S$ is necessary: for $A = \\Z[\\sqrt{-5}]$ and $S = A \\setminus \\set{0}$, $S^{-1}A$ is the field $\\Q(\\sqrt{-5})$, a UFD, while $A$ is not a UFD.\n\n## Explanations\n\n### L1 human\n\nTo prove that a Noetherian domain has unique factorization, it is enough to invert some prime elements and check unique factorization in the bigger ring.\nThis works because the inverted elements are prime, so they already factor well in $A$ and hide no failure of unique factorization.\nSo unique factorization can be tested in a simpler ring.\n\n### L2 human\n\nThe criterion is a standard way to show that rings given by generators and relations are UFDs.\nOne finds a prime element $p$ such that $A[1/p]$ is visibly factorial, often a polynomial ring or a Laurent polynomial ring, and concludes for $A$.\nIt recovers the fact that the [[polynomial ring over a UFD is a UFD|0D6KH8Y59N]] for Noetherian coefficient rings, as in the example with $\\Z[x]$.\nFor a [[Krull domain|CB9WJCN9BM]] $A$, the same idea shows that inverting prime elements does not change the [[divisor class group|WN8PK6RQZR]], since the height-one primes that meet $S$ are principal.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "06YMXJMXYS",
      "4C1AE84N5N"
    ],
    "kind": "statement",
    "msc": [
      "13F15",
      "13B30"
    ],
    "name": "Nagata's criterion for factoriality",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Unique_factorization_domain"
      }
    ],
    "tag": "GGEMJF449D",
    "uses": [
      "09S3PFBEPM",
      "0MVH4Y1CRZ",
      "0MNHXBSX3M",
      "AN9SKEZ3VY",
      "4C1AE84N5N",
      "06YMXJMXYS"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/GGEMJF449D/",
  "links": [
    {
      "field": "about",
      "target": "06YMXJMXYS"
    },
    {
      "field": "about",
      "target": "4C1AE84N5N"
    },
    {
      "field": "uses",
      "target": "09S3PFBEPM"
    },
    {
      "field": "uses",
      "target": "0MVH4Y1CRZ"
    },
    {
      "field": "uses",
      "target": "0MNHXBSX3M"
    },
    {
      "field": "uses",
      "target": "AN9SKEZ3VY"
    },
    {
      "field": "uses",
      "target": "4C1AE84N5N"
    },
    {
      "field": "uses",
      "target": "06YMXJMXYS"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GGEMJF449D.md",
  "msc": [
    "13F15",
    "13B30"
  ],
  "name": "Nagata's criterion for factoriality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GGE/GGEMJF449D.md",
  "status": null,
  "tag": "GGEMJF449D",
  "uri": "m2://GGEMJF449D",
  "url": "https://math2.tamnd.com/wiki/nagatas-criterion-for-factoriality/",
  "writing": {
    "tag": "GGEMJF449D",
    "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
      }
    ]
  }
}