On two problems concerning the Zariski topology of modules

Let R be an associative ring and let M be a left R-module. Let SpecR(M) be the collection of all prime submodules of M (equipped with classical Zariski topology). It is conjectured that every irreducible closed subset of SpecR(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 SpecR(M) is a spectral space.