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 In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, \dots, \textit, it can be an element of the unit interval ,1/math>. Von Ne ...
, 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 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 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, with the financial su ...
, series=Princeton Landmarks in Mathematics , isbn=978-0-691-05893-1 , mr=0120174 , year=1998
Ring theory