{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "052SDEPZH7"
    },
    {
      "field": "uses",
      "target": "053E86NWVK"
    },
    {
      "field": "uses",
      "target": "05THZKR9AN"
    },
    {
      "field": "uses",
      "target": "06RH4XAGRY"
    },
    {
      "field": "uses",
      "target": "0MNHXBSX3M"
    },
    {
      "field": "uses",
      "target": "0QA8CHH8MG"
    },
    {
      "field": "uses",
      "target": "0WZXK05ME7"
    }
  ],
  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nAn element $u \\in R$ is a unit of $R$ if there is an element $v \\in R$ such that $u \\cdot v = v \\cdot u = 1$.\nThe element $v$ is unique. It is the inverse of $u$, with the notation $u^{-1}$.\n\n## Remarks\n\nThe units of $\\mathbb{Z}$ are $1$ and $-1$.\nIn a [[field|0QA8CHH8MG]], every nonzero element is a unit.\nA real $n \\times n$ matrix is a unit of the matrix ring if and only if its determinant is not zero.\nThe units of $R$ form the [[group of units|05THZKR9AN]] of $R$.\n\n## Explanations\n\n### L1 human\n\nA unit is an element that you can divide by.\nIn the integers, you can divide by $1$ and $-1$ and stay in the integers, but you cannot divide by $2$.\nSo $1$ and $-1$ are the only units of $\\mathbb{Z}$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0XVEZDE9J8",
    "kind": "definition",
    "msc": [
      "16U60"
    ],
    "name": "Unit of a ring",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Unit_(ring_theory)"
      }
    ],
    "tag": "019NFGFPR5",
    "uses": [
      "0894J980RK"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/019NFGFPR5/",
  "links": [
    {
      "field": "defines",
      "target": "0XVEZDE9J8"
    },
    {
      "field": "uses",
      "target": "0894J980RK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/019NFGFPR5.md",
  "msc": [
    "16U60"
  ],
  "name": "Unit of a ring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/019/019NFGFPR5.md",
  "status": null,
  "tag": "019NFGFPR5",
  "uri": "m2://019NFGFPR5",
  "url": "https://math2.tamnd.com/wiki/unit-of-a-ring-definition/",
  "writing": {
    "tag": "019NFGFPR5",
    "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
      }
    ]
  }
}