On two problems concerning the Zariski topology of modules

Document Type: Research Paper

Authors

1 Department of pure Mathematics‎, ‎Faculty of mathematical Sciences‎, ‎University of Guilan‎, ‎P.O‎. ‎Box 41335-19141‎, ‎Rasht‎, ‎Iran.

2 Department of pure Mathematics‎, ‎Faculty of mathematical Sciences‎, ‎University of Guilan‎, ‎P‎. ‎O‎. ‎Box 41335-19141 Rasht‎, ‎Iran.

Abstract

Let $R$ be an associative ring and let $M$ be a left $R$-module. Let $Spec_{R}(M)$ be the collection of all prime submodules of  $M$ (equipped with classical Zariski topology). There is a conjecture  which says that every irreducible closed subset of $Spec_{R}(M)$ has a generic point. In this article we give an affirmative answer to this conjecture and show that if $M$ has a Noetherian spectrum, then $Spec_{R}(M)$ is a spectral space.

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H. Ansari-Toroghy and S. Keyvani, On the maximal spectrum of a module and Zariski topology, Bull. Malaysian Math. Soc. 21 (2015), no. 1, 303--316.

H. Ansari-Toroghy and R. Ovlyaee-Sarmazdeh, On the prime spectrum of X-injective modules, Comm. Algebra 38 (2010), no. 7, 2606--2621.

H. Ansari-Toroghy and R. Ovlyaee-Sarmazdeh, On the prime spectrum of a module and Zariski topologies, Comm. Algebra 38 (2010), no. 12, 4461--4475.

M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969

M. Behboodi and M. R. Haddadi, Clasical Zariski topology of modules and spectral spaces I, Int. Electron. J. Algebra 4 (2008) 104--130.

M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces II, Int. Electron. J. Algebra 4 (2008) 131--148.

N. Bourbaki, Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass, 1972.

M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969) 43--60.

C. P. Lu, Prime submodules of modules, Comment. Math. Univ. St. Paul 33 (1984), no. 1, 61--69.

C. P. Lu, Spectra of modules, Comm. Algebra 23 (1995), no. 10, 3741--3752.

C. P. Lu, Saturations of submodules, Comm. Algebra 31 (2003), no. 6, 2655--2673.

C. P. Lu, A module whose prime spectrum has the surjective natural map, Houston J. Math 33 (2007), no. 1, 125--143.

C. P. Lu, Modules with Noetherian spectrum, Comm. Algebra 38 (2010), no. 3, 807--828.

A. Marcelo, J. Masque, Prime submodules, the descent invariant, and modules of finite length, J. Algebra 189 (1997), no. 2, 273--293.

R. L. McCasland, M. E. Moore and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra 25 (1997), no. 1, 79--103.

J. Ohm and R. L. Pendleton, Ring with Noetherian spectrum, Duke. Math. J 35 (1968), no. 1, 631--639.

N. Van Sanh, L. P. Thao, N. F. A. Al-Meyahi, and K. P. Shum, Zariski topology of prime spectrum of a module, 461--477, Proceedings of the International Conference on Algebra, 2010.