{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "02HJYT53F7"
    },
    {
      "field": "uses",
      "target": "05WAF5T0GZ"
    },
    {
      "field": "uses",
      "target": "095ZHQ7WCJ"
    },
    {
      "field": "uses",
      "target": "0WD3SE7S1X"
    }
  ],
  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nAn [[ideal|0MYSA6MYNA]] $M$ of $R$ is maximal if it has these two properties.\n\n1. $M \\neq R$.\n2. For every ideal $I$ of $R$ with $M \\subseteq I$, either $I = M$ or $I = R$.\n\n## Remarks\n\nThe definition is the same for left, right and two-sided ideals, with ideals of the same type in condition 2.\nIn $\\mathbb{Z}$, the ideal $n\\mathbb{Z}$ with $n \\ge 0$ is maximal if and only if $n$ is a prime number.\nIn a [[commutative ring|0GYGHKX834]] $R$, an ideal $M$ is maximal if and only if [[$R/M$ is a field|0WD3SE7S1X]].\nThe zero ideal of a [[field|0QA8CHH8MG]] is its only maximal ideal.\n\n## Explanations\n\n### L1 human\n\nA maximal ideal is as large as possible without being the whole ring.\nIn the integers, $(6)$ is not maximal, because $(6) \\subseteq (2)$.\nThe ideal $(2)$ is maximal, because no ideal lies strictly between $(2)$ and $\\mathbb{Z}$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0PV74TR5FP",
    "kind": "definition",
    "msc": [
      "13A15"
    ],
    "name": "Maximal ideal",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Maximal_ideal"
      }
    ],
    "tag": "09J6KGT65G",
    "uses": [
      "0MYSA6MYNA"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/09J6KGT65G/",
  "links": [
    {
      "field": "defines",
      "target": "0PV74TR5FP"
    },
    {
      "field": "uses",
      "target": "0MYSA6MYNA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/09J6KGT65G.md",
  "msc": [
    "13A15"
  ],
  "name": "Maximal ideal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/09J/09J6KGT65G.md",
  "status": null,
  "tag": "09J6KGT65G",
  "uri": "m2://09J6KGT65G",
  "url": "https://math2.tamnd.com/wiki/maximal-ideal-definition/",
  "writing": {
    "tag": "09J6KGT65G",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}