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 e<sup>2</sup> = e, f<sup> 2</sup> = f, ef = fe = 0 then R(e + f ) = R(e) + R(f ).
References