In mathematics, a rank ring is a ring with a real-valued rank function behaving like the rank of an endomorphism. introduced rank rings in his work on
continuous geometry, and showed that the ring associated to a continuous geometry is a rank ring.
Definition
defined a ring to be a rank ring if it is
regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
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 ''e''
2 = ''e'', ''f''
2 = ''f'', ''ef'' = ''fe'' = 0 then ''R''(''e'' + ''f'') = ''R''(''e'') + ''R''(''f'').
References
*
*
*{{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
Princeton University Press is an independent Academic publishing, publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, ...
, series=Princeton Landmarks in Mathematics , isbn=978-0-691-05893-1 , mr=0120174 , year=1998
Ring theory