{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "MF89BJZ8KS"
    }
  ],
  "body": "\n## Summary\n\nAn algebra in a class of algebras is projective when every homomorphism from it into a quotient lifts through the quotient map.\nIt is the universal-algebra form of a projective object, with surjective homomorphisms in the role of epimorphisms.\nIn a variety, the projective algebras are exactly the retracts of free algebras.\nFor modules they are the [[projective modules|BFFA03GE3V]].\n\n## Explanations\n\n### L1 human\n\nA projective algebra can always be lifted back along an onto map.\nIf its image lands in a quotient, one can choose preimages for everything in a way that respects the operations.\nFree algebras are projective, because one only needs to choose a preimage for each free generator.\nThe group $\\Z/2\\Z$ is not projective among abelian groups: there is no way to lift the identity of $\\Z/2\\Z$ through $\\Z \\to \\Z/2\\Z$, since $\\Z$ has no element of order $2$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08B30",
      "18G05"
    ],
    "name": "Projective algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Projective_object"
      }
    ],
    "tag": "PZ41AE18X4"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/PZ41AE18X4/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/PZ41AE18X4.md",
  "msc": [
    "08B30",
    "18G05"
  ],
  "name": "Projective algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PZ4/PZ41AE18X4.md",
  "status": null,
  "tag": "PZ41AE18X4",
  "uri": "m2://PZ41AE18X4",
  "url": "https://tamnd.github.io/math2/wiki/projective-algebra/",
  "writing": {
    "tag": "PZ41AE18X4",
    "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
      }
    ]
  }
}