Rank ring

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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[edit]

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[edit]

  • 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, 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