Abstract
For any module M over a commutative ring R, \(Spec_{R}(M)\) (resp. \(Max_{R}(M)\)) of M is the collection of all prime (resp. maximal) submodules. In this article, we investigate the interplay between the topological properties of \(Max_{R}(M)\) and module theoretic properties of M. Also, for various types of modules M, we obtain some conditions under which \(Max_{R}(M)\) is homeomorphic with the maximal ideal space of some ring.
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References
Ansari-Toroghy, H., Ovlyaee-Sarmazdeh, R.: Modules for which the natural map of the maximal spectrum is surjective. Colloq. Math. 119, 217–227 (2010)
Ansari-Toroghy, H., Ovlyaee-Sarmazdeh, R.: On the prime spectrum of X-injective modules. Comm. Algebra 38, 2606–2621 (2010)
Ansari-Toroghy, H., Ovlyaee-Sarmazdeh, R.: On the prime spectrum of a module and Zariski topologies. Comm. Algebra 38, 4461–4475 (2010)
Ansari-Toroghy, H., Keyvani, S.: Some new classes of modules, (2011) (submitted)
Ansari-Toroghy, H., Keyvani, S.: Strongly top modules. Bull. Malays. Math. Sci. Soc. (2), 37(1): 73–82(2014)
Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley, Boston (1969)
Bourbaki, N.: Algebra commutative, Chap. 1,2. Hermann, Paris (1961)
Chin-Pi, Lu: Prime submodules of modules. Comment. Math. Univ. St. Pauli 33(1), 61–69 (1984)
Chin-Pi, Lu: Spectra of modules. Comm. Algebra 23(10), 3741–3752 (1995)
Chin-Pi, Lu: The Zariski topology on the prime spectrum of a module. Houston J. Math. 25(3), 417–432 (1999)
Chin-Pi, Lu: A module whose prime spectrum has the surjective natural map. Houston J. Math. 33(1), 125–143 (2007)
Hochster, M.: Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142, 43–60 (1969)
McCasland, R.L., Moore, M.E., Smith, P.F.: On the spectrum of a module over a commutative ring. Comm. Algebra 25(1), 79–103 (1997)
Moller, Jesper M.: General Topology, Matematisk Institut, Universitetsparken 5, DK-2100 Kobenhavn
Ohm, J., Pendleton, R.L.: Ring with noetherian spectrum. Duke Math. J. 35(3), 631–639 (1968)
Shirazi, M.M., Sharif, H.: Hilbert modules. Int. J. Pure Appl. Math. 20(1), 1–7 (2005)
Acknowledgments
The authors would like to thank Dr. R. Ovlyaee-Sarmazdeh for several helpful conversations in this work. Also we are grateful to the referee for careful reading of the manuscript.
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Communicated by Siamak Yassemi.
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Ansari-Toroghy, H., Keyvani, S. On the Maximal Spectrum of a Module and Zariski Topology. Bull. Malays. Math. Sci. Soc. 38, 303–316 (2015). https://doi.org/10.1007/s40840-014-0020-1
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DOI: https://doi.org/10.1007/s40840-014-0020-1