{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nLet $R$ be an [[integral domain|09S3PFBEPM]] and let $p \\in R$ be a [[prime element|0MNHXBSX3M]].\n\n1. By the definition of a prime element, $p \\neq 0$ and $p$ is not a [[unit|019NFGFPR5]]. These are the first two conditions for $p$ to be [[irreducible|052SDEPZH7]].\n2. Let $a, b \\in R$ with $p = a b$. Then $p$ [[divides|04GNK7K4VP]] $a b$, because $a b = p \\cdot 1$.\n3. Since $p$ is prime, $p \\mid a$ or $p \\mid b$. The roles of $a$ and $b$ are symmetric, so we may assume $p \\mid a$.\n4. Write $a = p c$ with $c \\in R$. Then $p \\cdot 1 = p = a b = p (c b)$.\n5. Since $p \\neq 0$, [[cancellation|0GJM24EXP9]] gives $c b = 1$. Since $R$ is commutative, also $b c = 1$, so $b$ is a unit.\n6. Thus in every factorization $p = a b$, one factor is a unit. With step 1, $p$ is irreducible.\n\n## Idea\n\nIf $p = a b$, then $p$ divides the product $a b$, so $p$ divides one factor, say $a = p c$. Then $p = p c b$, and cancelling $p$ shows that $b$ is a unit.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "13A05"
    ],
    "name": "Proof that prime elements are irreducible in an integral domain",
    "proves": "0F0K8DB9TD",
    "sources": [
      {
        "url": "https://proofwiki.org/wiki/Prime_Element_of_Integral_Domain_is_Irreducible"
      }
    ],
    "tag": "V4JCKJNVYR",
    "uses": [
      "09S3PFBEPM",
      "0MNHXBSX3M",
      "052SDEPZH7",
      "04GNK7K4VP",
      "019NFGFPR5",
      "0GJM24EXP9"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/V4JCKJNVYR/",
  "links": [
    {
      "field": "proves",
      "target": "0F0K8DB9TD"
    },
    {
      "field": "uses",
      "target": "09S3PFBEPM"
    },
    {
      "field": "uses",
      "target": "0MNHXBSX3M"
    },
    {
      "field": "uses",
      "target": "052SDEPZH7"
    },
    {
      "field": "uses",
      "target": "04GNK7K4VP"
    },
    {
      "field": "uses",
      "target": "019NFGFPR5"
    },
    {
      "field": "uses",
      "target": "0GJM24EXP9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/V4JCKJNVYR.md",
  "msc": [
    "13A05"
  ],
  "name": "Proof that prime elements are irreducible in an integral domain",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V4J/V4JCKJNVYR.md",
  "status": null,
  "tag": "V4JCKJNVYR",
  "uri": "m2://V4JCKJNVYR",
  "url": "https://math2.tamnd.com/wiki/proof-that-prime-elements-are-irreducible-in-an-integral-domain/",
  "writing": {
    "tag": "V4JCKJNVYR",
    "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
      }
    ]
  }
}