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