{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $L$ be a [[complemented|ZT47BK9Y6K]] [[modular lattice|QX7WZ92QAK]] with least element $0$ and greatest element $1$.\nElements $a, b \\in L$ are *perspective* if some $x \\in L$ satisfies $a \\wedge x = b \\wedge x = 0$ and $a \\vee x = b \\vee x$.\nElements $a_1, \\dots, a_n \\in L$ are *independent* if $(a_1 \\vee \\cdots \\vee a_k) \\wedge a_{k+1} = 0$ for each $k$ with $1 \\le k \u003c n$.\nA *homogeneous basis* of $L$ is a family $a_1, \\dots, a_n$ of independent elements with $a_1 \\vee \\cdots \\vee a_n = 1$.\nIn addition, $a_i$ and $a_j$ must be perspective for all $i, j$.\n\nSuppose that $L$ has a homogeneous basis with $n \\ge 4$ elements.\nThen there is a [[von Neumann regular ring|DXX7GNY3HE]] $R$ such that $L$ is isomorphic to the lattice of principal right ideals of $R$.\nHere the principal right ideals are ordered by inclusion.\n\n## Remarks\n\nFor a regular ring $R$, every principal right ideal is generated by an idempotent, and the principal right ideals form a complemented modular lattice; the theorem gives the converse.\nThe ring is built from $L$ much as a division ring is built from a projective space of dimension at least $3$.\nThe bound $n \\ge 4$ plays the role of that dimension bound.\n\nExample: for a division ring $D$ and $n \\ge 4$, the lattice of subspaces of $D^n$ has the homogeneous basis of coordinate lines.\nIt is isomorphic to the lattice of principal right ideals of the matrix ring $M_n(D)$, which is regular.\n\nVon Neumann applied the theorem to [[continuous geometries|5CET1KWWWH]], where it yields continuous regular rings.\n\n## Explanations\n\n### L1 human\n\nClassical projective geometry shows that a projective space of dimension at least $3$ always comes from a division ring, so its points and lines can be given coordinates.\nVon Neumann's theorem does the same for complemented modular lattices with a large enough homogeneous basis.\nThe coordinates now live in a regular ring, a ring in which every element $a$ has some $x$ with $axa = a$.\n\n### L2 human\n\nThe theorem identifies complemented modular lattices that have a homogeneous basis with at least $4$ elements with lattices of principal right ideals of regular rings.\nIt explains why regular rings are the natural algebraic counterpart of complemented modular lattices.\nIts main application is to [[continuous geometries|5CET1KWWWH]], whose coordinate rings are the continuous regular rings, including rings built from type II$_1$ factors.\nThe independence and perspectivity conditions are lattice versions of a direct sum decomposition into isomorphic summands, like $D^n = D \\oplus \\cdots \\oplus D$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "QX7WZ92QAK",
      "DXX7GNY3HE"
    ],
    "kind": "statement",
    "msc": [
      "06C20",
      "16E50"
    ],
    "name": "von Neumann coordinatization theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Continuous_geometry"
      }
    ],
    "tag": "VDRCDZ048J",
    "uses": [
      "QX7WZ92QAK",
      "ZT47BK9Y6K",
      "DXX7GNY3HE"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/VDRCDZ048J/",
  "links": [
    {
      "field": "about",
      "target": "QX7WZ92QAK"
    },
    {
      "field": "about",
      "target": "DXX7GNY3HE"
    },
    {
      "field": "uses",
      "target": "QX7WZ92QAK"
    },
    {
      "field": "uses",
      "target": "ZT47BK9Y6K"
    },
    {
      "field": "uses",
      "target": "DXX7GNY3HE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/VDRCDZ048J.md",
  "msc": [
    "06C20",
    "16E50"
  ],
  "name": "von Neumann coordinatization theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VDR/VDRCDZ048J.md",
  "status": null,
  "tag": "VDRCDZ048J",
  "uri": "m2://VDRCDZ048J",
  "url": "https://tamnd.github.io/math2/wiki/von-neumann-coordinatization-theorem/",
  "writing": {
    "tag": "VDRCDZ048J",
    "kind": "statement",
    "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
      }
    ]
  }
}