{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\n$R$ is *reduced* if it has no nonzero [[nilpotent elements|0N9R5NN6TW]].\nThat is, for all $r \\in R$ and all integers $n \\ge 1$, if $r^n = 0$ then $r = 0$.\n\n## Remarks\n\nExample: $\\Z/6\\Z$ is reduced; if $r^n \\equiv 0 \\pmod 6$, then $2$ and $3$ divide $r$, so $r \\equiv 0$.\nNon-example: $\\Z/4\\Z$ is not reduced, since $2^2 = 0$ and $2 \\ne 0$.\nEquivalent forms: the [[nilradical|QQDN6EYQNJ]] of $R$ is zero; the zero ideal is a [[radical ideal|S4GQN5V54X]]; for all $r \\in R$, $r^2 = 0$ implies $r = 0$.\nEvery [[integral domain|09S3PFBEPM]] is reduced, and so is every product of integral domains.\nThe same condition defines reduced rings that are not commutative.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "SZHY975FFX",
    "kind": "definition",
    "msc": [
      "13A99",
      "13A15"
    ],
    "name": "Reduced ring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Reduced_ring"
      }
    ],
    "tag": "KXTQ10RDSX",
    "uses": [
      "0GYGHKX834",
      "0N9R5NN6TW"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/KXTQ10RDSX/",
  "links": [
    {
      "field": "defines",
      "target": "SZHY975FFX"
    },
    {
      "field": "uses",
      "target": "0GYGHKX834"
    },
    {
      "field": "uses",
      "target": "0N9R5NN6TW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/KXTQ10RDSX.md",
  "msc": [
    "13A99",
    "13A15"
  ],
  "name": "Reduced ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KXT/KXTQ10RDSX.md",
  "status": null,
  "tag": "KXTQ10RDSX",
  "uri": "m2://KXTQ10RDSX",
  "url": "https://math2.tamnd.com/wiki/reduced-ring-definition/",
  "writing": {
    "tag": "KXTQ10RDSX",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}