{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0VFW4Y69BT"
    }
  ],
  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nA two-sided ideal of $R$ is a subset $I \\subseteq R$ with these properties.\n\n1. $I$ is a subgroup of the additive group $(R, +)$.\n2. For all $r \\in R$ and $a \\in I$, $r a \\in I$ and $a r \\in I$.\n\nSo a two-sided ideal is a [[left ideal|0MYSA6MYNA]] that is also a right ideal.\n\n## Remarks\n\nIn a [[commutative ring|0GYGHKX834]], left ideals, right ideals and two-sided ideals are the same.\nThe [[kernel|0AAHE7GEW6]] of every [[ring homomorphism|0GGE0Z0JQ1]] is a two-sided ideal.\nThe [[quotient ring|0VFW4Y69BT]] $R/I$ exists exactly when $I$ is a two-sided ideal.\nIn the ring of $n \\times n$ matrices over a field, the only two-sided ideals are $\\{0\\}$ and the whole ring.\n\n## Explanations\n\n### L1 human\n\nIn a noncommutative ring, $r a$ and $a r$ can be different.\nA two-sided ideal must absorb products from both sides.\nThis is the condition that makes the quotient ring work.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0CD87SFM2N",
    "kind": "definition",
    "msc": [
      "16D25"
    ],
    "name": "Two-sided ideal",
    "sources": [
      {
        "locator": "Definitions",
        "url": "https://en.wikipedia.org/wiki/Ideal_(ring_theory)"
      }
    ],
    "tag": "0DDA6CY7GJ",
    "uses": [
      "0MYSA6MYNA"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0DDA6CY7GJ/",
  "links": [
    {
      "field": "defines",
      "target": "0CD87SFM2N"
    },
    {
      "field": "uses",
      "target": "0MYSA6MYNA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0DDA6CY7GJ.md",
  "msc": [
    "16D25"
  ],
  "name": "Two-sided ideal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0DD/0DDA6CY7GJ.md",
  "status": null,
  "tag": "0DDA6CY7GJ",
  "uri": "m2://0DDA6CY7GJ",
  "url": "https://math2.tamnd.com/wiki/two-sided-ideal-definition/",
  "writing": {
    "tag": "0DDA6CY7GJ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}