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Rank Ring
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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor. Definition Menger and Birkhoff gave axioms for projective geometry in terms of the lattice of linear subspaces of projective space. Von Neumann's axioms for continuous geometry are a weakened form of these axioms. A continuous geometry is a lattice ''L'' with the following properties *''L'' is modular. *''L'' is complete. *The lattice operations ∧, ∨ satisfy a certain continuity property, *:(\bigwedge_ a_\alpha)\lor b = \bigwedge_\alpha (a_\alpha\lor b), where ''A'' is a directe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Regular Ring
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the element ''a;'' in general ''x'' is not uniquely determined by ''a''. Von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left ''R''-module is flat. Von Neumann regular rings were introduced by under the name of "regular rings", in the course of his study of von Neumann algebras and continuous geometry. Von Neumann regular rings should not be confused with the unrelated regular rings and regular local rings of commutative algebra. An element ''a'' of a ring is called a von Neumann regular element if there exists an ''x'' such that .Kaplansky (1972) p.110 An ideal \mathfrak is called a (von Neumann) regular ideal if for every element ''a'' in \mathfrak there exists an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Journal Of Mathematics
The ''Canadian Journal of Mathematics'' (french: Journal canadien de mathématiques) is a bimonthly mathematics journal published by the Canadian Mathematical Society. It was established in 1949 by H. S. M. Coxeter and G. de B. Robinson. The current editors-in-chief of the journal are Louigi Addario-Berry and Eyal Goren. The journal publishes articles in all areas of mathematics. See also * Canadian Mathematical Bulletin The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antoni ... References External links * University of Toronto Press academic journals Mathematics journals Publications established in 1949 Bimonthly journals Multilingual journals Cambridge University Press academic journals Academic journals associated with learned and professional societies of Canada ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |