In mathematics, a rank ring is a ring with a real-valued rank function behaving like the rank of an endomorphism. introduced rank rings in his work on continuous geometry

Continuous geometry

In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]...

, and showed that the ring associated to a continuous geometry is a rank ring.

Definition

defined a ring to be a rank ring if it is regular

Von Neumann regular ring

In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R withOne may think of x as a "weak inverse" of a...

and has a real-valued rank function R with the following properties:

• 0 ≤ R(a) ≤ 1 for all a
• R(a) = 0 if and only if a = 0
• R(1) = 1
• R(ab) ≤ R(a), R(ab) ≤ R(b)
• If e² = e, f² = f, ef = fe = 0 then R(e + f) = R(e) + R(f).