%0 Journal Article %T On two problems concerning the Zariski topology of modules %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A Ansari-Toroghy, H. %A Ovlyaee-Sarmazdeh, R. %A Pourmortazavi, Seyed Sajad %D 2016 %\ 08/01/2016 %V 42 %N 4 %P 941-948 %! On two problems concerning the Zariski topology of modules %K Prime spectrum %K classical Zariski topology %K spectral space %R %X 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. %U http://bims.iranjournals.ir/article_846_1769c60f5d1f171166300f224d7e2683.pdf