{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "8MGH8PGJWR"
    }
  ],
  "body": "\n## Summary\n\nThe total ring of fractions of a [[commutative ring|0310REJJY8]] $R$ is the [[localization|4C1AE84N5N]] of $R$ at the set of elements that are not [[zero divisors|0NNRDW10RZ]].\nIt is the largest localization in which $R$ still embeds.\nFor an [[integral domain|0QVD7WBFGN]], it is the [[field of fractions|0F2PTGX1A2]].\n\n## Explanations\n\n### L1 human\n\nTo build fractions from a ring, you want to divide by as many elements as possible.\nYou cannot divide by a zero divisor $s$, with $sa = 0$ for some $a \\ne 0$, without making $a$ equal to zero.\nSo the safe choice is to allow every element that is not a zero divisor as a denominator.\nFor the integers this gives the rational numbers.\nThe [[definition|8MGH8PGJWR]] states it.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13B30"
    ],
    "name": "Total ring of fractions",
    "same_as": {
      "wikidata": "Q5695923",
      "wikipedia": "Total ring of fractions"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Total_ring_of_fractions"
      }
    ],
    "specializes": [
      "4C1AE84N5N"
    ],
    "tag": "K5Z67HR9ET"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/K5Z67HR9ET/",
  "links": [
    {
      "field": "specializes",
      "target": "4C1AE84N5N"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/K5Z67HR9ET.md",
  "msc": [
    "13B30"
  ],
  "name": "Total ring of fractions",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/K5Z/K5Z67HR9ET.md",
  "status": null,
  "tag": "K5Z67HR9ET",
  "uri": "m2://K5Z67HR9ET",
  "url": "https://math2.tamnd.com/wiki/total-ring-of-fractions/",
  "writing": {
    "tag": "K5Z67HR9ET",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}