# Rank ring

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In [mathematics](/source/mathematics), a '''rank ring''' is a [ring](/source/ring_(mathematics)) with a [real](/source/real_number)-valued rank function behaving like the rank of an [endomorphism](/source/endomorphism). {{harvs|first=John|last=von Neumann|authorlink=John von Neumann|year=1998|txt=yes}} introduced rank rings in his work on [continuous geometry](/source/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](/source/von_Neumann_regular_ring) 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''<sup>2</sup>&nbsp;=&nbsp;''e'', ''f''<sup>&thinsp;2</sup>&nbsp;=&nbsp;''f'', ''ef''&nbsp;=&nbsp;''fe''&nbsp;=&nbsp;0 then ''R''(''e''&nbsp;+&nbsp;''f''&thinsp;)&nbsp;=&nbsp;''R''(''e'')&nbsp;+&nbsp;''R''(''f''&thinsp;).

==References==

*{{Citation | last1=Halperin | first1=Israel | title=Regular rank rings | doi=10.4153/CJM-1965-071-4 | mr=0191926 | year=1965 | journal=[Canadian Journal of Mathematics](/source/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 | orig-date=1960 | url=https://books.google.com/books?id=onE5HncE-HgC | publisher=[Princeton University Press](/source/Princeton_University_Press) | series=Princeton Landmarks in Mathematics | isbn=978-0-691-05893-1 | mr=0120174 | year=1998}}

Category:Ring theory

{{Algebra-stub}}

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