Rank ring

In mathematics, a rank ring is a ring with a real-valued rank function behaving like the rank of an endomorphism. John von Neumann (1998) introduced rank rings in his work on continuous geometry, and showed that the ring associated to a continuous geometry is a rank ring.

Definition

John von Neumann (1998, p.231) 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, f² = f, ef = fe = 0 then R(e + f ) = R(e) + R(f ).

References

• Halperin, Israel (1965), "Regular rank rings", Canadian Journal of Mathematics, 17: 709–719, doi:10.4153/CJM-1965-071-4, ISSN 0008-414X, MR 0191926
• von Neumann, John (1936), "Examples of continuous geometries.", Proc. Natl. Acad. Sci. USA, 22 (2): 101–108, Bibcode:1936PNAS...22..101N, doi:10.1073/pnas.22.2.101, JFM 62.0648.03, JSTOR 86391, PMC 1076713, PMID 16588050
• von Neumann, John (1998) [1960], Continuous geometry, Princeton Landmarks in Mathematics, Princeton University Press, ISBN 978-0-691-05893-1, MR 0120174