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http://dx.doi.org/10.4134/JKMS.j180649

SOME ASPECTS OF ZARISKI TOPOLOGY FOR MULTIPLICATION MODULES AND THEIR ATTACHED FRAMES AND QUANTALES  

Castro, Jaime (Escuela de Ingenieria y Ciencias Instituto Tecnologico y de Estudios Superiores de Monterrey)
Rios, Jose (Instituto de Matematicas Universidad Nacional Autonoma de Mexico)
Tapia, Gustavo (Instituto de Ingenieria y Tecnologia Universidad Autonoma de Ciudad Juarez)
Publication Information
Journal of the Korean Mathematical Society / v.56, no.5, 2019 , pp. 1285-1307 More about this Journal
Abstract
For a multiplication R-module M we consider the Zariski topology in the set Spec (M) of prime submodules of M. We investigate the relationship between the algebraic properties of the submodules of M and the topological properties of some subspaces of Spec (M). We also consider some topological aspects of certain frames. We prove that if R is a commutative ring and M is a multiplication R-module, then the lattice Semp (M/N) of semiprime submodules of M/N is a spatial frame for every submodule N of M. When M is a quasi projective module, we obtain that the interval ${\uparrow}(N)^{Semp}(M)=\{P{\in}Semp(M){\mid}N{\subseteq}P\}$ and the lattice Semp (M/N) are isomorphic as frames. Finally, we obtain results about quantales and the classical Krull dimension of M.
Keywords
Zariski topology; multiplication modules; frames; spatial frames; quantales; dense subspaces; irreducible subspaces; Krull dimension;
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1 J. Abuhlail and C. Lomp, On topological lattices and their applications to module theory, J. Algebra Appl. 15 (2016), no. 3, 1650046, 21 pp. https://doi.org/10.1142/S0219498816500468   DOI
2 T. Albu, Sur la dimension de Gabriel des modules, in Seminar, F., Pareigis, K.-B., eds. Algebra Beriche. Vol. 21. Munchen: Verlag Uni-Druck, 1974.
3 D. D. Anderson, Some remarks on multiplication ideals, Math. Japon. 25 (1980), no. 4, 463-469.
4 D. D. Anderson, Some remarks on multiplication ideals. II, Comm. Algebra 28 (2000), no. 5, 2577-2583. https://doi.org/10.1080/00927870008826980   DOI
5 M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, MA, 1969.
6 J. A. Beachy, M-injective modules and prime M-ideals, Comm. Algebra 30 (2002), no. 10, 4649-4676. https://doi.org/10.1081/AGB-120014660   DOI
7 L. Bican, P. Jambor, T. Kepka, and P. Nemec, Prime and coprime modules, Fund. Math. 107 (1980), no. 1, 33-45. https://doi.org/10.4064/fm-107-1-33-45   DOI
8 N. Bourbaki, Commutative Algebra, Berlin, Springer-Verlag, 1998.
9 J. Dauns, Prime modules, J. Reine Angew. Math. 298 (1978), 156-181. https://doi.org/10.1515/crll.1978.298.156
10 Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra 16 (1988), no. 4, 755-779. https://doi.org/10.1080/00927878808823601   DOI
11 A. Jawad, A Zariski topology for modules, Comm. Algebra. 39 (2011), no. 11, 4749-4768.
12 P. T. Johnstone, Stone Spaces, reprint of the 1982 edition, Cambridge Studies in Advanced Mathematics, 3, Cambridge University Press, Cambridge, 1986.
13 D. Lu and W. Yu, On prime spectrums of commutative rings, Comm. Algebra 34 (2006), no. 7, 2667-2672. https://doi.org/10.1080/00927870600651612   DOI
14 M. G. Medina-Barcenas, L. Morales-Callejas, L. Sandoval, and A. Zaldivar, Attaching topological spaces to a module (I) : Sobriety and spatiality, J. Pure Appl. Algebra 222 (2018), no. 5, 1026-1048. https://doi.org/10.1016/j.jpaa.2017.06.005   DOI
15 M. G. Medina-Barcenas, L. A. Zaldivar-Corichi, and M. L. S. Sandoval-Miranda, A generalization of quantales with applications to modules and rings, J. Pure Appl. Algebra 220 (2016), no. 5, 1837-1857. https://doi.org/10.1016/j.jpaa.2015.10.004   DOI
16 A. Barnard, Multiplication modules, J. Algebra 71 (1981), no. 1, 174-178. https://doi.org/10.1016/0021-8693(81)90112-5   DOI
17 A. G. Naoum, On the ring of endomorphisms of a finitely generated multiplication module, Period. Math. Hungar. 21 (1990), no. 3, 249-255. https://doi.org/10.1007/BF02651092   DOI
18 J. C. Perez and J. Rios, Prime submodules and local Gabriel correspondence in ${\sigma}$ [M], Comm. Algebra 40 (2012), no. 1, 213-232. https://doi.org/10.1080/00927872.2010.529095   DOI
19 F. Raggi, J. Rios, H. Rincon, R. Fernandez-Alonso, and C. Signoret, C. Prime and irreducible preradicals, J. Algebra Appl. 4 (2005), no. 4, 451-466. https://doi.org/10.1142/S0219498805001290   DOI
20 J. Picado and A. Pultr, Frames and Locales, Frontiers in Mathematics, Birkhauser/Springer Basel AG, Basel, 2012. https://doi.org/10.1007/978-3-0348-0154-6
21 F. Raggi, J. Rios, H. Rincon, R. Fernandez-Alonso, and C. Signoret, Semiprime preradicals, Comm. Algebra 37 (2009), no. 8, 2811-2822. https://doi.org/10.1080/00927870802623476   DOI
22 K. I. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics Series, 234, Longman Scientific & Technical, Harlow, 1990.
23 N. Schwartz and M. Tressl, Elementary properties of minimal and maximal points in Zariski spectra, J. Algebra 323 (2010), no. 3, 698-728. https://doi.org/10.1016/j.jalgebra.2009.11.003   DOI
24 P. F. Smith, Some remarks on multiplication modules, Arch. Math. (Basel) 50 (1988), no. 3, 223-235. https://doi.org/10.1007/BF01187738   DOI
25 B. Stenstrom, Rings of Quotients, Graduate Texts in Mathematics, New York: Springer-Verlag, 1975.
26 J. C. Perez, M. Medina, J. Rios, and A. Zaldivar, On semiprime Goldie modules, Comm. Algebra 44 (2016), no. 11, 4749-4768. https://doi.org/10.1080/00927872.2015.1113290   DOI
27 A. A. Tuganbaev, Multiplication modules, J. Math. Sci. (N. Y.) 123 (2004), no. 2, 3839-3905. https://doi.org/10.1023/B:JOTH.0000036653.76231.05   DOI
28 I. E. Wijayanti and R. Wisbauer, On coprime modules and comodules, Comm. Algebra 37 (2009), no. 4, 1308-1333. https://doi.org/10.1080/00927870802466926   DOI
29 R. Wisbauer, Modules and algebras: bimodule structure and group actions on algebras, Pitman Monographs and Surveys in Pure and Applied Mathematics, 81, Longman, Harlow, 1996.
30 R. Wisbauer, Foundations of Module and Ring Theory, revised and translated from the 1988 German edition, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
31 G. Zhang, W. Tong, and F. Wang, Spectrum of a noncommutative ring, Comm. Algebra 34 (2006), no. 8, 2795-2810. https://doi.org/10.1080/00927870600636936   DOI