# 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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### History

• Receive Date: 18 December 2014
• Revise Date: 13 June 2015
• Accept Date: 13 June 2015