{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "07QY5RPFBR"
    },
    {
      "field": "uses",
      "target": "0HJ4DTW6FN"
    },
    {
      "field": "uses",
      "target": "0K5W69AC0J"
    }
  ],
  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nA subring of $R$ is a subset $S \\subseteq R$ with these properties.\n\n1. $1 \\in S$.\n2. If $a, b \\in S$, then $a - b \\in S$.\n3. If $a, b \\in S$, then $a \\cdot b \\in S$.\n\nA subring $S$ is a ring with the addition and the multiplication of $R$, restricted to $S$.\n\n## Remarks\n\nThe integers $\\mathbb{Z}$ are a subring of $\\mathbb{Q}$, and $\\mathbb{Q}$ is a subring of $\\mathbb{R}$.\nThe set $2\\mathbb{Z}$ of even integers is closed under addition and multiplication, but it is not a subring of $\\mathbb{Z}$, because $1 \\notin 2\\mathbb{Z}$.\nAuthors who do not require a ring to have an identity also do not require $1 \\in S$.\n\n## Explanations\n\n### L1 human\n\nA subring is a smaller ring inside a ring, with the same operations.\nFor example, the integers sit inside the rational numbers.\nThe sum, the difference and the product of two integers are again integers.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0GNCRPYXYV",
    "kind": "definition",
    "msc": [
      "16B99"
    ],
    "name": "Subring",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Subring"
      }
    ],
    "tag": "0859D58WPD",
    "uses": [
      "0894J980RK"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0859D58WPD/",
  "links": [
    {
      "field": "defines",
      "target": "0GNCRPYXYV"
    },
    {
      "field": "uses",
      "target": "0894J980RK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0859D58WPD.md",
  "msc": [
    "16B99"
  ],
  "name": "Subring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/085/0859D58WPD.md",
  "status": null,
  "tag": "0859D58WPD",
  "uri": "m2://0859D58WPD",
  "url": "https://math2.tamnd.com/wiki/subring-definition/",
  "writing": {
    "tag": "0859D58WPD",
    "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
      }
    ]
  }
}