{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "09N7DZ7SDB"
    }
  ],
  "body": "\n## Statement\n\nLet $f \\colon R \\to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].\nThen $f$ is injective if and only if its [[kernel|0AAHE7GEW6]] is $\\{0\\}$.\n\n## Remarks\n\nThe proof uses only the addition: $f(a) = f(b)$ holds if and only if $f(a - b) = 0$, that is, $a - b \\in \\ker f$.\nThe same criterion holds for group homomorphisms and for linear maps.\n\n## Explanations\n\n### L1 human\n\nIf two elements have the same image, then their difference goes to $0$.\nSo when only $0$ goes to $0$, no two different elements have the same image.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "16B99"
    ],
    "name": "A ring homomorphism is injective iff its kernel is zero",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Ring_homomorphism"
      }
    ],
    "tag": "0MMCB8RXGA",
    "uses": [
      "0GGE0Z0JQ1",
      "0AAHE7GEW6"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0MMCB8RXGA/",
  "links": [
    {
      "field": "uses",
      "target": "0GGE0Z0JQ1"
    },
    {
      "field": "uses",
      "target": "0AAHE7GEW6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0MMCB8RXGA.md",
  "msc": [
    "16B99"
  ],
  "name": "A ring homomorphism is injective iff its kernel is zero",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0MM/0MMCB8RXGA.md",
  "status": null,
  "tag": "0MMCB8RXGA",
  "uri": "m2://0MMCB8RXGA",
  "url": "https://math2.tamnd.com/wiki/a-ring-homomorphism-is-injective-iff-its-kernel-is-zero/",
  "writing": {
    "tag": "0MMCB8RXGA",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}