Non-stable K theory of regular rings
Thesis title in Czech: | |
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Thesis title in English: | Non-stable K theory of regular rings |
English key words: | K -theory, von Neuman regular ring, refinement monoid, countable, |
Academic year of topic announcement: | 2011/2012 |
Thesis type: | dissertation |
Thesis language: | angličtina |
Department: | Department of Algebra (32-KA) |
Supervisor: | doc. Mgr. Pavel Růžička, Ph.D. |
Author: | hidden - assigned and confirmed by the Study Dept. |
Date of registration: | 26.09.2012 |
Date of assignment: | 26.09.2012 |
Confirmed by Study dept. on: | 08.11.2012 |
Guidelines |
Let R be a ring. The monoid of projective modules, V(R), is the set of isomorphism classes of finitely generated projective left R-modules with the opeartion corresponding to the direct sum of the modules. For a von Neumann regular ring R, V(R) is a refinement conical monoid. In general, the problem of the realization of a refinement monoid as the monoid V(R) for a regular ring R has a negative answer due to F. Wehrung. However the problem remains open for countable refinement monoids. This is connected to the seperativity problem, in particular, to the question whether every von Neumann regular ring is separative. A positive answer to the above realization problem would reject the separativity conjecture.
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References |
[1] P. Ara, The realization problem for von Neumann regular rings. “RING THEORY 2007
Proceedings of the Fifth China–Japan–Korea Conference Tokyo, Japan, 10 – 15 September 2007”, 2008, 316 pp. [2] P. Ara, The regular algebra of a poset. Trans. Amer. Math. Soc. 362 (2010), no. 3, 1505-1546. [3] P. Ara, M. Brustenga, The regular algebra of a quiver. J. Algebra, 309 (2007), 207-535. [4] K. R. Goodearl, “Von Neumann Regular Rings”. Pitman, London, 1979. xvii + 369 pp. [5] K. R. Goodearl, “Partially Ordered Abelian Groups with Interpolation (Mathematical Surveys and Monographs)”. AMS, 1986; xxii + 336 pp. [6] K. R. Goodearl, Von Neumann regular rings and direct sum decomposition problems. Abelian Groups and Modules, Kluwer, Dordrecht, 1995, 249–255. [7] J. Moncasi, A regular ring whose K0 is not a Riesz group. Comm. Algebra 13 (1985), no. 1, 125-131. [8] F. Wehrung, Non measurability properties of interpolation vector spaces. Israel Journal of Mathematics 103, no. 1 (1998), 177–206. [9] F. Wehrung, Coordinatization of lattices by regular rings without unit and Banaschewski functions. Algebra Universalis 64 (2010), no. 1-2, 49–67. |