{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "B1Q5F5VNSK"
    },
    {
      "field": "uses",
      "target": "MM3TQ1WJWF"
    },
    {
      "field": "uses",
      "target": "QHD4KJYFSA"
    },
    {
      "field": "uses",
      "target": "QSK2J5HEER"
    },
    {
      "field": "uses",
      "target": "RT2D9JEVRR"
    },
    {
      "field": "uses",
      "target": "TCF1PXZ5YN"
    },
    {
      "field": "uses",
      "target": "VW2H7873YM"
    },
    {
      "field": "about",
      "target": "QHD4KJYFSA"
    },
    {
      "field": "about",
      "target": "QSK2J5HEER"
    },
    {
      "field": "defines",
      "target": "WKG2DZDEBT"
    }
  ],
  "body": "\n## Summary\n\nA nil ideal of a [[ring|0B1261B5ED]] is a one-sided or two-sided ideal all of whose elements are [[nilpotent|0ZB2JG1M9C]].\nNil ideals are a main object of radical theory, the study of the parts of a ring that are discarded to reach simpler rings.\nIn a commutative ring the largest nil ideal is the [[nilradical|QQDN6EYQNJ]].\nWhether every nil one-sided ideal lies in a nil two-sided ideal is the open [[Köthe conjecture|RT2D9JEVRR]].\n\n## Explanations\n\n### L1 human\n\nAn element is nilpotent when some power of it is zero.\nA nil ideal is an ideal in which every element has this property, though the power needed may depend on the element.\nIn the ring $\\Z/8\\Z$, the ideal of even residues is nil: every even residue $a$ satisfies $a^3 = 0$.\nIn $\\Z$, the ideal $2\\Z$ is not nil, since no power of $2$ is zero.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "16N40"
    ],
    "name": "Nil ideal",
    "same_as": {
      "wikidata": "Q453275",
      "wikipedia": "Nil ideal"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Nil_ideal"
      }
    ],
    "tag": "NZ1G9WRWFB"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/NZ1G9WRWFB/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/NZ1G9WRWFB.md",
  "msc": [
    "16N40"
  ],
  "name": "Nil ideal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NZ1/NZ1G9WRWFB.md",
  "status": null,
  "tag": "NZ1G9WRWFB",
  "uri": "m2://NZ1G9WRWFB",
  "url": "https://tamnd.github.io/math2/wiki/nil-ideal/",
  "writing": {
    "tag": "NZ1G9WRWFB",
    "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
      }
    ]
  }
}