Rank ring

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, 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 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, f&thinsp;2 = f, ef = fe = 0 then R(e + f&thinsp;) = R(e) + R(f&thinsp;).