rank ring
In mathematics, a rank ring is a ring with a real-valued rank function behaving like the rank of an endomorphism. {{harvs|first=John|last=von Neumann|authorlink=John von Neumann|year=1998|txt=yes}} introduced rank rings in his work on continuous geometry, and showed that the ring associated to a continuous geometry is a rank ring.
Definition
{{harvs|txt|first=John|last=von Neumann|year=1998|loc=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
- {{Citation | last1=Halperin | first1=Israel | title=Regular rank rings | url=http://cms.math.ca/10.4153/CJM-1965-071-4 | doi=10.4153/CJM-1965-071-4 | mr=0191926 | year=1965 | journal=Canadian Journal of Mathematics | issn=0008-414X | volume=17 | pages=709–719| doi-access=free }}
- {{Citation | last1=von Neumann | first1=John | author1-link=John von Neumann | title=Examples of continuous geometries. | jstor=86391 | doi=10.1073/pnas.22.2.101 | jfm=62.0648.03 | year=1936 | journal=Proc. Natl. Acad. Sci. USA | volume=22 | issue=2 | pages=101–108 | pmid=16588050 | pmc=1076713| bibcode=1936PNAS...22..101N | doi-access=free }}
- {{Citation | last1=von Neumann | first1=John | author1-link=John von Neumann | title=Continuous geometry | origyear=1960 | url=https://books.google.com/books?id=onE5HncE-HgC | publisher=Princeton University Press | series=Princeton Landmarks in Mathematics | isbn=978-0-691-05893-1 | mr=0120174 | year=1998}}
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