{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "3Z7YMK2WDS"
    },
    {
      "field": "defines",
      "target": "PYABBA1QQE"
    }
  ],
  "body": "\n## Summary\n\nA universally catenary ring is a [[Noetherian ring|0670T59HGE]] whose finitely generated algebras are all [[catenary|B2CYR292J6]].\nBeing catenary is not preserved when one adjoins variables, so this stronger condition is the one that behaves well under the usual constructions.\nFields, $\\Z$, Cohen–Macaulay rings and [[excellent rings|YF7YFQ58QJ]] are universally catenary.\n\n## Explanations\n\n### L1 human\n\nA catenary ring is one where all maximal chains of prime ideals between two primes have the same length.\nA universally catenary ring keeps this property after any finite number of polynomial variables are added and after any quotient is taken.\nFor example, every finitely generated algebra over a field, such as $k[x, y]/(xy)$, is catenary because the field $k$ is universally catenary.\nThe [[definition|PYABBA1QQE]] makes this precise.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13C15"
    ],
    "name": "Universally catenary ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Catenary_ring"
      }
    ],
    "specializes": [
      "B2CYR292J6"
    ],
    "tag": "5WAFPTH09V"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/5WAFPTH09V/",
  "links": [
    {
      "field": "specializes",
      "target": "B2CYR292J6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/5WAFPTH09V.md",
  "msc": [
    "13C15"
  ],
  "name": "Universally catenary ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5WA/5WAFPTH09V.md",
  "status": null,
  "tag": "5WAFPTH09V",
  "uri": "m2://5WAFPTH09V",
  "url": "https://math2.tamnd.com/wiki/universally-catenary-ring/",
  "writing": {
    "tag": "5WAFPTH09V",
    "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
      }
    ]
  }
}