{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "C7X657TF19"
    },
    {
      "field": "about",
      "target": "HE7MP4HP6Q"
    },
    {
      "field": "defines",
      "target": "ZV04WWYE2Y"
    }
  ],
  "body": "\n## Summary\n\nAn Artinian ring is a ring in which every decreasing chain of ideals stops after finitely many steps.\nFor commutative rings, Artinian rings are exactly the [[Noetherian rings|0670T59HGE]] in which every prime ideal is maximal, by the [[Akizuki–Hopkins–Levitzki theorem|C7X657TF19]].\nBy the [[structure theorem for Artinian rings|HE7MP4HP6Q]], a commutative Artinian ring is a finite product of Artinian local rings.\nFinite rings, fields and finite-dimensional algebras over a field are Artinian.\n\n## Explanations\n\n### L1 human\n\nIn an Artinian ring you cannot keep shrinking ideals forever: every decreasing chain becomes constant.\nSuch rings are small in a precise sense, like finite-dimensional algebras.\nFor example, $\\Z/12\\Z$ has only six ideals, so any decreasing chain of ideals must stop.\nThe ring $\\Z$ is not Artinian, because $(2) \\supset (4) \\supset (8) \\supset \\cdots$ never stops.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13E10",
      "16P20"
    ],
    "name": "Artinian ring",
    "same_as": {
      "wikidata": "Q713084",
      "wikipedia": "Artinian ring"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Artinian_ring"
      }
    ],
    "specializes": [
      "0310REJJY8"
    ],
    "tag": "20QEAM8AX2"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/20QEAM8AX2/",
  "links": [
    {
      "field": "specializes",
      "target": "0310REJJY8"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/20QEAM8AX2.md",
  "msc": [
    "13E10",
    "16P20"
  ],
  "name": "Artinian ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/20Q/20QEAM8AX2.md",
  "status": null,
  "tag": "20QEAM8AX2",
  "uri": "m2://20QEAM8AX2",
  "url": "https://math2.tamnd.com/wiki/artinian-ring/",
  "writing": {
    "tag": "20QEAM8AX2",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}