{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0AWMPX35CW"
    }
  ],
  "body": "\n## Statement\n\nLet $R$ be a [[principal ideal domain|0RZYDVZ475]] and let $p \\in R$ with $p \\neq 0$.\nThen $p$ is [[irreducible|052SDEPZH7]] if and only if the principal ideal $(p)$ is a [[maximal ideal|09J6KGT65G]].\n\n## Remarks\n\nThe proof: an ideal $I$ with $(p) \\subseteq I$ is $(a)$ with $a \\mid p$. If $p$ is irreducible, then $a$ is a unit or an associate of $p$, so $I = R$ or $I = (p)$.\nA consequence is that $R/(p)$ is a [[field|0QA8CHH8MG]] for each irreducible $p$. For example, $\\mathbb{Q}[x]/(x^2 - 2)$ is a field.\nIn a principal ideal domain, every nonzero prime ideal is maximal.\n\n## Explanations\n\n### L1 human\n\nIn a principal ideal domain, ideals that contain $(p)$ come from the divisors of $p$.\nAn irreducible $p$ has only trivial divisors.\nSo no ideal fits strictly between $(p)$ and the whole ring.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "13F10"
    ],
    "name": "In a PID a nonzero element is irreducible iff it generates a maximal ideal",
    "role": "theorem",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Principal_ideal_domain"
      }
    ],
    "tag": "095ZHQ7WCJ",
    "uses": [
      "0RZYDVZ475",
      "052SDEPZH7",
      "09J6KGT65G",
      "00A2EDFT9Q"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/095ZHQ7WCJ/",
  "links": [
    {
      "field": "uses",
      "target": "0RZYDVZ475"
    },
    {
      "field": "uses",
      "target": "052SDEPZH7"
    },
    {
      "field": "uses",
      "target": "09J6KGT65G"
    },
    {
      "field": "uses",
      "target": "00A2EDFT9Q"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/095ZHQ7WCJ.md",
  "msc": [
    "13F10"
  ],
  "name": "In a PID a nonzero element is irreducible iff it generates a maximal ideal",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/095/095ZHQ7WCJ.md",
  "status": null,
  "tag": "095ZHQ7WCJ",
  "uri": "m2://095ZHQ7WCJ",
  "url": "https://math2.tamnd.com/wiki/in-a-pid-a-nonzero-element-is-irreducible-iff-it-generates-a-maximal-ideal/",
  "writing": {
    "tag": "095ZHQ7WCJ",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}