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, http://cms.math.ca/10.4153/CJM-1965-071-4
* von Neumann, John (1936), "Examples of continuous geometries.", Proc. Natl. Acad. Sci. USA 22 (2): 101–108, doi:10.1073/pnas.22.2.101, PMID 16588050, Bibcode: 1936PNAS...22..101N
* von Neumann, John (1998) [1960], Continuous geometry, Princeton Landmarks in Mathematics, Princeton University Press, ISBN 978-0-691-05893-1, https://books.google.com/books?id=onE5HncE-HgC

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Rank ring. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Rank_ring&oldid=64402"