{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "JEMH9P1NVX"
    },
    {
      "field": "about",
      "target": "W27Z60Q1PF"
    },
    {
      "field": "uses",
      "target": "W27Z60Q1PF"
    }
  ],
  "body": "\n## Summary\n\nA normal ring is a commutative ring whose [[localization|X6JD4AG973]] at every prime ideal is an [[integrally closed domain|H20P600HQS]].\nIt is the local form of integral closedness, and it allows rings that are not domains, such as products of integrally closed domains.\nA Noetherian normal ring is a finite product of integrally closed domains.\nNormal rings are the rings of functions on normal schemes, and normality is described by [[Serre's criterion|W27Z60Q1PF]].\n\n## Explanations\n\n### L1 human\n\nBeing integrally closed is a property of a domain, but a space can have several pieces, and its ring of functions is then not a domain.\nNormality asks for integral closedness near each point, by looking at the ring of germs of functions there.\nThe ring $\\Z \\times \\Z$ is normal, as it describes two disjoint copies of the same well-behaved space.\nThe ring $k[x, y]/(y^2 - x^3)$ of the cusp curve is not normal, because it fails at the cusp.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13B22"
    ],
    "name": "Normal ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Normal_scheme"
      }
    ],
    "tag": "KR7SWBYHRQ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/KR7SWBYHRQ/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/KR7SWBYHRQ.md",
  "msc": [
    "13B22"
  ],
  "name": "Normal ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KR7/KR7SWBYHRQ.md",
  "status": null,
  "tag": "KR7SWBYHRQ",
  "uri": "m2://KR7SWBYHRQ",
  "url": "https://math2.tamnd.com/wiki/normal-ring/",
  "writing": {
    "tag": "KR7SWBYHRQ",
    "kind": "concept",
    "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
      }
    ]
  }
}