102 research outputs found
Torsion-freeness and non-singularity over right p.p.-rings
AbstractA right R-module M is non-singular if xIâ 0 for all non-zero xâM and all essential right ideals I of R. The module M is torsion-free if Tor1R(M,R/Rr)=0 for all râR. This paper shows that, for a ring R, the classes of torsion-free and non-singular right R-modules coincide if and only if R is a right Utumi-p.p.-ring with no infinite set of orthogonal idempotents. Several examples and applications of this result are presented. Special emphasis is given to the case where the maximal right ring of quotients of R is a perfect left localization of R
Geometry of Reidemeister classes and twisted Burnside theorem
This is a (mostly expository) paper on Reidemeister classes, twisted
Burnside-Frobenius theory, congruences, R-infinity property and all that. It
was written in 2005 and published in 2008. We post it as it was, only the
bibliography data is updated. For some of the recent progress see e.g.
arXiv:0903.4533, arXiv:0903.3455, arXiv:0802.2937, arXiv:0712.2601,
arXiv:0704.3411, arXiv:math/0703744, arXiv:math/0606725, arXiv:math/0606764,
arXiv:0805.1371 and references there
The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach
We compare two influential ways of defining a generalized notion of space.
The first, inspired by Gelfand duality, states that the category of
'noncommutative spaces' is the opposite of the category of C*-algebras. The
second, loosely generalizing Stone duality, maintains that the category of
'pointfree spaces' is the opposite of the category of frames (i.e., complete
lattices in which the meet distributes over arbitrary joins). One possible
relationship between these two notions of space was unearthed by Banaschewski
and Mulvey, who proved a constructive version of Gelfand duality in which the
Gelfand spectrum of a commutative C*-algebra comes out as a pointfree space.
Being constructive, this result applies in arbitrary toposes (with natural
numbers objects, so that internal C*-algebras can be defined). Earlier work by
the first three authors, shows how a noncommutative C*-algebra gives rise to a
commutative one internal to a certain sheaf topos. The latter, then, has a
constructive Gelfand spectrum, also internal to the topos in question. After a
brief review of this work, we compute the so-called external description of
this internal spectrum, which in principle is a fibered pointfree space in the
familiar topos Sets of sets and functions. However, we obtain the external
spectrum as a fibered topological space in the usual sense. This leads to an
explicit Gelfand transform, as well as to a topological reinterpretation of the
Kochen-Specker Theorem of quantum mechanics, which supplements the remarkable
topos-theoretic version of this theorem due to Butterfield and Isham.Comment: 12 page
Zariski topology on the spectrum of graded classical prime submodules
[EN] Let R be a G-graded commutative ring with identity and let M be a graded R-module. A proper graded submodule N of M is called graded classical prime if for every a, b Âż h(R), m Âż h(M), whenever abm Âż N, then either am Âż N or bm Âż N. The spectrum of graded classical prime submodules of M is denoted by Cl.Specg(M). We topologize Cl.Specg (M) with the quasi-Zariski topology, which is analogous to that for Specg(R).Yousefian Darani, A.; Motmaen, S. (2013). Zariski topology on the spectrum of graded classical prime submodules. Applied General Topology. 14(2):159-169. doi:10.4995/agt.2013.1586.SWORD159169142S. Ebrahimi Atani and F. Farzalipour, On weakly prime submodules, Tamkang Journal of Mathematics 38, no. 3 (2007), 247-252.S. Ebrahimi Atani and F. Farzalipour, On graded multiplication modules, Chiang-Mai Journal of Science, to appear.S. Ebrahimi Atani and F.E.K. Saraei, Graded modules which satisfy the Gr-Radical formola, Thai Journal of Mathematics 8, no. 1 (2010), 161-170.P. Lu, The Zariski topology on the prime spectrum of a module, Houston J. Math. 25, no. 3 (1999), 417-425.McCasland, R. L., Moore, M. E., & Smith, P. F. (1997). On the spectrum of a module over a commutative ring. Communications in Algebra, 25(1), 79-103. doi:10.1080/00927879708825840K. H. Oral, U. Tekir and A.G. Agargun, On graded prime and primary submodules, Turk. J. Math. 25, no. 3 (1999), 417-425.Roberts, P. C. (1998). Multiplicities and Chern Classes in Local Algebra. doi:10.1017/cbo9780511529986Sharp, R. Y. (1986). Asymptotic Behaviour of Certain Sets of Attached Prime Ideals. Journal of the London Mathematical Society, s2-34(2), 212-218. doi:10.1112/jlms/s2-34.2.212BAZIAR, M., & BEHBOODI, M. (2009). CLASSICAL PRIMARY SUBMODULES AND DECOMPOSITION THEORY OF MODULES. Journal of Algebra and Its Applications, 08(03), 351-362. doi:10.1142/s0219498809003369M. Behboodi and H. Koohi, Weakly prime modules, Vietnam J. Math. 32, no. 2 (2004), 185â195.M. Behboodi and M. J. Noori, Zariski-Like topology on the classical prime spectrum of a module, Bull. Iranian Math. Soc. 35, no. 1 (2009), 255â271.M. Behboodi and S. H. Shojaee, On chains of classical prime submodules and dimension theory of modules, Bulletin of the Iranian Mathematical Society 36 (2010), 149â166.J. Dauns, Prime modules, J. Reine Angew. Math. 298 (1978), 156â181.S. Ebrahimi Atani, On graded prime submodules, Chiang Mai J. Sci. 33, no. 1 (2006), 3â7
Domains of commutative C*-subalgebras
A C*-algebra is determined to a great extent by the partial order of its
commutative C*-algebras. We study order-theoretic properties of this dcpo. Many
properties coincide: the dcpo is, equivalently, algebraic, continuous,
meet-continuous, atomistic, quasi-algebraic, or quasi-continuous, if and only
if the C*-algebra is scattered. For C*-algebras with enough projections, these
properties are equivalent to finite-dimensionality. Approximately
finite-dimensional elements of the dcpo correspond to Boolean subalgebras of
the projections of the C*-algebra, which determine the projections up to
isomorphism. Scattered C*-algebras are finite-dimensional if and only if their
dcpo is Lawson-scattered. General C*-algebras are finite-dimensional if and
only if their dcpo is order-scattered.Comment: 42 page
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