{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]], let $S$ be a nonzero [[ring|0894J980RK]] and let $f \\colon K \\to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].\nThen $f$ is injective.\n\n## Remarks\n\nThe proof: if $a \\neq 0$ and $f(a) = 0$, then $1 = f(1) = f(a) f(a^{-1}) = 0$ in $S$. This contradicts $S \\neq 0$.\nThe hypothesis $S \\neq 0$ is necessary, because the map from $K$ to the [[zero ring|008TBCYDPX]] is a ring homomorphism.\nA consequence: a homomorphism of fields makes the first field a copy of a subfield of the second.\n\n## Explanations\n\n### L1 human\n\nA field has no ideal other than $0$ and the whole field.\nThe kernel of $f$ is an ideal, and it does not contain $1$.\nSo the kernel is $0$, and $f$ is injective.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "12E99"
    ],
    "name": "Ring homomorphisms from a field are injective",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Ring_homomorphism"
      }
    ],
    "tag": "09N7DZ7SDB",
    "uses": [
      "0GGE0Z0JQ1",
      "0QA8CHH8MG",
      "0MMCB8RXGA"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/09N7DZ7SDB/",
  "links": [
    {
      "field": "uses",
      "target": "0GGE0Z0JQ1"
    },
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    },
    {
      "field": "uses",
      "target": "0MMCB8RXGA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/09N7DZ7SDB.md",
  "msc": [
    "12E99"
  ],
  "name": "Ring homomorphisms from a field are injective",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/09N/09N7DZ7SDB.md",
  "status": null,
  "tag": "09N7DZ7SDB",
  "uri": "m2://09N7DZ7SDB",
  "url": "https://math2.tamnd.com/wiki/ring-homomorphisms-from-a-field-are-injective/",
  "writing": {
    "tag": "09N7DZ7SDB",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}