{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "10TJ890QK0"
    },
    {
      "field": "uses",
      "target": "3F2W916W64"
    },
    {
      "field": "uses",
      "target": "6EWEMZ6FX1"
    },
    {
      "field": "uses",
      "target": "84XXPH6XJ8"
    },
    {
      "field": "uses",
      "target": "AAV343QP5E"
    },
    {
      "field": "uses",
      "target": "CZF5696TPY"
    },
    {
      "field": "uses",
      "target": "JG7WB75BHC"
    },
    {
      "field": "uses",
      "target": "N783XNJNBV"
    },
    {
      "field": "uses",
      "target": "WV3PBEK76J"
    },
    {
      "field": "uses",
      "target": "Y50YJMKRAM"
    },
    {
      "field": "about",
      "target": "JG7WB75BHC"
    },
    {
      "field": "about",
      "target": "N783XNJNBV"
    }
  ],
  "body": "\n## Summary\n\nA quotient algebra is the algebra of classes of a [[congruence relation|DCSXXT4SER]], with the operations computed on representatives.\nIt generalizes the [[quotient group|0KYX2N69J0]] and the [[quotient ring|0076HWHGHQ]].\nEvery homomorphic image of an algebra is isomorphic to one of its quotients, by the [[homomorphism theorem|N783XNJNBV]].\n\n## Explanations\n\n### L1 human\n\nStart with an algebra and a congruence, which says which elements to treat as equal.\nGlue each class of equal elements into a single point.\nTo apply an operation to some classes, pick one element from each class, apply the operation, and take the class of the result.\nThe congruence guarantees that the choice does not matter.\nFor the integers with congruence modulo $5$, this gives the five-element ring $\\Z/5\\Z$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08A30"
    ],
    "name": "Quotient algebra",
    "same_as": {
      "wikidata": "Q2589508",
      "wikipedia": "Quotient (universal algebra)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quotient_(universal_algebra)"
      }
    ],
    "tag": "N5KE7FH6CX"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/N5KE7FH6CX/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/N5KE7FH6CX.md",
  "msc": [
    "08A30"
  ],
  "name": "Quotient algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/N5K/N5KE7FH6CX.md",
  "status": null,
  "tag": "N5KE7FH6CX",
  "uri": "m2://N5KE7FH6CX",
  "url": "https://tamnd.github.io/math2/wiki/quotient-algebra/",
  "writing": {
    "tag": "N5KE7FH6CX",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}