{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1X1Y2AW4B0"
    },
    {
      "field": "uses",
      "target": "3CZ8PXR9JY"
    },
    {
      "field": "defines",
      "target": "2K5GKVZR78"
    },
    {
      "field": "about",
      "target": "3CZ8PXR9JY"
    }
  ],
  "body": "\n## Summary\n\nThe nilradical of a commutative ring is the set of its [[nilpotent elements|0N9R5NN6TW]].\nIt is an ideal, the [[radical|437YRCNYRJ]] of the zero ideal.\nIt equals the [[intersection of all prime ideals|3CZ8PXR9JY]] of the ring, and a ring with zero nilradical is called [[reduced|SZHY975FFX]].\n\n## Explanations\n\n### L1 human\n\nAn element is nilpotent when some power of it is zero.\nThe nilradical collects all such elements.\nIn $\\Z/12\\Z$, the nilpotent elements are $0$ and $6$, since $6^2 = 36 = 0$ in $\\Z/12\\Z$, while for instance no power of $2$ or $4$ is divisible by $3$.\nIn $\\Z$ or in any field the nilradical is $\\set{0}$.\nGeometrically, nilpotent functions vanish at every point, so the nilradical is invisible to the points of the [[spectrum|FB45BHHA2J]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13A15"
    ],
    "name": "Nilradical",
    "same_as": {
      "wikidata": "Q3341881",
      "wikipedia": "Nilradical of a ring"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Nilradical_of_a_ring"
      }
    ],
    "tag": "QQDN6EYQNJ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/QQDN6EYQNJ/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/QQDN6EYQNJ.md",
  "msc": [
    "13A15"
  ],
  "name": "Nilradical",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QQD/QQDN6EYQNJ.md",
  "status": null,
  "tag": "QQDN6EYQNJ",
  "uri": "m2://QQDN6EYQNJ",
  "url": "https://math2.tamnd.com/wiki/nilradical/",
  "writing": {
    "tag": "QQDN6EYQNJ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}