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, f 2 = 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
Stub icon

This algebra-related article is a stub. You can help Wikipedia by adding missing information.

Content Disclaimer

Informasi ini disarikan dari Wikipedia dan disajikan kembali untuk tujuan edukasi. Konten tersedia di bawah lisensi CC BY-SA 3.0. Kami tidak bertanggung jawab atas ketidakakuratan data yang bersumber dari kontribusi publik tersebut.

  1. The information displayed on this website is sourced in part or in whole from Wikipedia and has been adapted for the purpose of restating it. We strive to provide accurate and relevant information, however:
  2. There is no guarantee of absolute accuracy. Wikipedia is an open, collaborative project that can be edited by anyone, so information is subject to change.
  3. It is not intended to constitute professional advice. The content displayed is for informational and educational purposes only. For important decisions (e.g., medical, legal, or financial), please consult a professional.
  4. Content copyright. Wikipedia is licensed under the Creative Commons Attribution-ShareAlike License (CC BY-SA). This means that content may be reused with appropriate attribution and shared under a similar license.
  5. Responsible use. Any risk arising from the use of information from this website is entirely the responsibility of the user.