{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nLet $R$ be a [[principal ideal domain|0RZYDVZ475]] and let $p \\in R$ with $p \\neq 0$.\n\n1. For $b \\in R$, the set $b R = \\set{b t : t \\in R}$ is an ideal that contains $b$. Every ideal containing $b$ contains $b R$. So the [[ideal generated by|030TZ0DQWT]] $b$ is $(b) = b R$.\n2. For $b \\in R$, $(b) = R$ exactly when $b$ is a [[unit|019NFGFPR5]]. If $(b) = R$, then $1 = b t$ for some $t \\in R$, so $b$ is a unit. If $b t = 1$, then each $r \\in R$ equals $b (t r) \\in b R$.\n\nFirst suppose that $p$ is [[irreducible|052SDEPZH7]].\n\n3. Since $p$ is not a unit, step 2 gives $(p) \\neq R$.\n4. Let $I$ be an ideal with $(p) \\subseteq I$. Since $R$ is a principal ideal domain, $I = (a)$ for some $a \\in R$, a [[principal ideal|00A2EDFT9Q]].\n5. Since $p \\in (a) = a R$, there is $b \\in R$ with $p = a b$. Since $p$ is irreducible, $a$ is a unit or $b$ is a unit.\n6. If $a$ is a unit, then $I = (a) = R$ by step 2.\n7. If $b$ is a unit, then $a = p b^{-1} \\in (p)$. So $(a) \\subseteq (p)$, and $I = (p)$.\n8. By steps 3 to 7, $(p)$ is a [[maximal ideal|09J6KGT65G]].\n\nNow suppose that $(p)$ is a maximal ideal.\n\n9. We have $p \\neq 0$ by hypothesis. Since $(p) \\neq R$, step 2 shows that $p$ is not a unit.\n10. Let $p = a b$ with $a, b \\in R$. Then $p \\in (a)$, so $(p) \\subseteq (a)$. By maximality, $(a) = (p)$ or $(a) = R$.\n11. If $(a) = R$, then $a$ is a unit by step 2.\n12. If $(a) = (p)$, then $a = p t$ for some $t \\in R$. So $p \\cdot 1 = p = a b = p (t b)$.\n13. Since $p \\neq 0$, [[cancellation|0GJM24EXP9]] gives $t b = 1$, so $b$ is a unit.\n14. So in every factorization $p = a b$, one factor is a unit. With step 9, $p$ is irreducible.\n\n## Idea\n\nIn a principal ideal domain, an ideal containing $(p)$ has the form $(a)$ with $a$ a divisor of $p$. So the ideals between $(p)$ and $R$ match the divisors of $p$. The element $p$ has only trivial divisors exactly when there are no ideals strictly between $(p)$ and $R$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "13F10"
    ],
    "name": "Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal",
    "proves": "095ZHQ7WCJ",
    "sources": [
      {
        "url": "https://proofwiki.org/wiki/Principal_Ideal_of_Principal_Ideal_Domain_is_of_Irreducible_Element_iff_Maximal"
      }
    ],
    "tag": "VBJVQ0CDBM",
    "uses": [
      "0RZYDVZ475",
      "052SDEPZH7",
      "09J6KGT65G",
      "00A2EDFT9Q",
      "030TZ0DQWT",
      "019NFGFPR5",
      "0GJM24EXP9"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/VBJVQ0CDBM/",
  "links": [
    {
      "field": "proves",
      "target": "095ZHQ7WCJ"
    },
    {
      "field": "uses",
      "target": "0RZYDVZ475"
    },
    {
      "field": "uses",
      "target": "052SDEPZH7"
    },
    {
      "field": "uses",
      "target": "09J6KGT65G"
    },
    {
      "field": "uses",
      "target": "00A2EDFT9Q"
    },
    {
      "field": "uses",
      "target": "030TZ0DQWT"
    },
    {
      "field": "uses",
      "target": "019NFGFPR5"
    },
    {
      "field": "uses",
      "target": "0GJM24EXP9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/VBJVQ0CDBM.md",
  "msc": [
    "13F10"
  ],
  "name": "Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VBJ/VBJVQ0CDBM.md",
  "status": null,
  "tag": "VBJVQ0CDBM",
  "uri": "m2://VBJVQ0CDBM",
  "url": "https://math2.tamnd.com/wiki/proof-that-in-a-pid-a-nonzero-element-is-irreducible-iff-it-generates-a-maximal-ideal/",
  "writing": {
    "tag": "VBJVQ0CDBM",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}