{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nAn [[ideal|0MYSA6MYNA]] $I$ of $R$ is *irreducible* if these two conditions hold:\n\n1. $I \\ne R$.\n2. For all ideals $J$ and $K$ of $R$ with $I = J \\cap K$, one has $I = J$ or $I = K$.\n\n## Remarks\n\nExample: every prime ideal $P$ is irreducible; if $P = J \\cap K$ with $a \\in J \\setminus P$ and $b \\in K \\setminus P$, then $ab \\in J \\cap K = P$, a contradiction.\nNon-example: $(6)$ in $\\Z$ fails clause 2, since $(6) = (2) \\cap (3)$ with both terms strictly larger.\nIn a Noetherian ring, every irreducible ideal is [[primary|V026AAV8E6]].\nThe converse fails: in $k[x, y]$ over a field $k$, the ideal $(x^2, xy, y^2)$ is primary, but it equals $(x^2, y) \\cap (x, y^2)$.\nSome authors call an ideal with clause 2 alone meet-irreducible, and then allow $I = R$.\n\n## Explanations\n\n### L1 human\n\nAn irreducible ideal is a proper ideal that is not the intersection of two strictly larger ideals.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "7PDWB550C9",
    "kind": "definition",
    "msc": [
      "13A15"
    ],
    "name": "Irreducible ideal",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Irreducible_ideal"
      }
    ],
    "tag": "CAKG0X11TJ",
    "uses": [
      "0GYGHKX834",
      "0MYSA6MYNA"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/CAKG0X11TJ/",
  "links": [
    {
      "field": "defines",
      "target": "7PDWB550C9"
    },
    {
      "field": "uses",
      "target": "0GYGHKX834"
    },
    {
      "field": "uses",
      "target": "0MYSA6MYNA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/CAKG0X11TJ.md",
  "msc": [
    "13A15"
  ],
  "name": "Irreducible ideal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CAK/CAKG0X11TJ.md",
  "status": null,
  "tag": "CAKG0X11TJ",
  "uri": "m2://CAKG0X11TJ",
  "url": "https://math2.tamnd.com/wiki/irreducible-ideal-definition/",
  "writing": {
    "tag": "CAKG0X11TJ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}