{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "08C1NWBNHA"
    },
    {
      "field": "uses",
      "target": "0CHE3J7QMX"
    },
    {
      "field": "uses",
      "target": "0F0K8DB9TD"
    },
    {
      "field": "uses",
      "target": "0QRCFZKXH8"
    }
  ],
  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nAn element $p \\in R$ is prime if it has these three properties.\n\n1. $p \\neq 0$.\n2. $p$ is not a [[unit|019NFGFPR5]].\n3. For all $a, b \\in R$, if $p \\mid a b$, then $p \\mid a$ or $p \\mid b$.\n\n## Remarks\n\nA nonzero $p$ is prime if and only if the principal ideal $(p)$ is a [[prime ideal|06HC6HK0Z8]].\nThe prime elements of $\\mathbb{Z}$ are the numbers $p$ and $-p$ with $p$ a prime number.\nIn an integral domain, [[prime elements are irreducible|0F0K8DB9TD]].\n\n## Explanations\n\n### L1 human\n\nEuclid's lemma says that a prime number dividing a product divides one of the factors.\nA prime element of a ring is an element with this property.\nIn the integers, $6$ is not prime: $6$ divides $2 \\cdot 3$, but not $2$ or $3$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "01VH21EV6F",
    "kind": "definition",
    "msc": [
      "13A05"
    ],
    "name": "Prime element",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Prime_element"
      }
    ],
    "tag": "0MNHXBSX3M",
    "uses": [
      "04GNK7K4VP",
      "019NFGFPR5"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0MNHXBSX3M/",
  "links": [
    {
      "field": "defines",
      "target": "01VH21EV6F"
    },
    {
      "field": "uses",
      "target": "04GNK7K4VP"
    },
    {
      "field": "uses",
      "target": "019NFGFPR5"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0MNHXBSX3M.md",
  "msc": [
    "13A05"
  ],
  "name": "Prime element",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0MN/0MNHXBSX3M.md",
  "status": null,
  "tag": "0MNHXBSX3M",
  "uri": "m2://0MNHXBSX3M",
  "url": "https://math2.tamnd.com/wiki/prime-element-definition/",
  "writing": {
    "tag": "0MNHXBSX3M",
    "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
      }
    ]
  }
}