---
tag: VDRCDZ048J
kind: statement
role: theorem
name: von Neumann coordinatization theorem
msc: [06C20, "16E50"]
uses: [QX7WZ92QAK, ZT47BK9Y6K, DXX7GNY3HE]
about: [QX7WZ92QAK, DXX7GNY3HE]
sources:
  - url: https://en.wikipedia.org/wiki/Continuous_geometry
---

## Statement

Let $L$ be a [[complemented|ZT47BK9Y6K]] [[modular lattice|QX7WZ92QAK]] with least element $0$ and greatest element $1$.
Elements $a, b \in L$ are *perspective* if some $x \in L$ satisfies $a \wedge x = b \wedge x = 0$ and $a \vee x = b \vee x$.
Elements $a_1, \dots, a_n \in L$ are *independent* if $(a_1 \vee \cdots \vee a_k) \wedge a_{k+1} = 0$ for each $k$ with $1 \le k < n$.
A *homogeneous basis* of $L$ is a family $a_1, \dots, a_n$ of independent elements with $a_1 \vee \cdots \vee a_n = 1$.
In addition, $a_i$ and $a_j$ must be perspective for all $i, j$.

Suppose that $L$ has a homogeneous basis with $n \ge 4$ elements.
Then there is a [[von Neumann regular ring|DXX7GNY3HE]] $R$ such that $L$ is isomorphic to the lattice of principal right ideals of $R$.
Here the principal right ideals are ordered by inclusion.

## Remarks

For 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.
The ring is built from $L$ much as a division ring is built from a projective space of dimension at least $3$.
The bound $n \ge 4$ plays the role of that dimension bound.

Example: for a division ring $D$ and $n \ge 4$, the lattice of subspaces of $D^n$ has the homogeneous basis of coordinate lines.
It is isomorphic to the lattice of principal right ideals of the matrix ring $M_n(D)$, which is regular.

Von Neumann applied the theorem to [[continuous geometries|5CET1KWWWH]], where it yields continuous regular rings.

## Explanations

### L1 human

Classical 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.
Von Neumann's theorem does the same for complemented modular lattices with a large enough homogeneous basis.
The coordinates now live in a regular ring, a ring in which every element $a$ has some $x$ with $axa = a$.

### L2 human

The theorem identifies complemented modular lattices that have a homogeneous basis with at least $4$ elements with lattices of principal right ideals of regular rings.
It explains why regular rings are the natural algebraic counterpart of complemented modular lattices.
Its main application is to [[continuous geometries|5CET1KWWWH]], whose coordinate rings are the continuous regular rings, including rings built from type II$_1$ factors.
The independence and perspectivity conditions are lattice versions of a direct sum decomposition into isomorphic summands, like $D^n = D \oplus \cdots \oplus D$.