{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "1QS5A6XNXA"
    }
  ],
  "body": "\n## Summary\n\nThe direct limit of rings glues a directed family of [[rings|0B1261B5ED]] $R_i$, connected by ring homomorphisms $R_i \\to R_j$ for $i \\le j$, into one ring.\nThis ring receives a map from each $R_i$.\nIt is the [[direct limit|X9HEQ29Q49]] taken in the category of rings, and it is computed on the underlying sets.\nEvery element of the limit comes from some $R_i$, and two elements are equal if they become equal in some later $R_k$.\nPolynomial rings in infinitely many variables and many infinite-dimensional algebras arise this way.\n\n## Explanations\n\n### L1 human\n\nImagine a chain of rings, each sitting inside the next one, such as $k[x_1] \\subseteq k[x_1, x_2] \\subseteq k[x_1, x_2, x_3] \\subseteq \\cdots$.\nThe direct limit is the union of the chain, here the polynomial ring in infinitely many variables.\nWhen the maps are not inclusions, elements of different rings are identified as soon as they become equal further along.\nEach computation in the limit takes place in one of the rings of the family, since only finitely many elements are involved.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "16S10",
      "18A30",
      "13B02"
    ],
    "name": "Direct limit of rings",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Direct_limit"
      }
    ],
    "tag": "Q5T05ET56S"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/Q5T05ET56S/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/Q5T05ET56S.md",
  "msc": [
    "16S10",
    "18A30",
    "13B02"
  ],
  "name": "Direct limit of rings",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Q5T/Q5T05ET56S.md",
  "status": null,
  "tag": "Q5T05ET56S",
  "uri": "m2://Q5T05ET56S",
  "url": "https://tamnd.github.io/math2/wiki/direct-limit-of-rings/",
  "writing": {
    "tag": "Q5T05ET56S",
    "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
      }
    ]
  }
}