Attaching topological spaces to a module (I): Sobriety and spatiality

In this paper we study some frames associated to an R-module M. We define semiprimitive submodules and we prove that they form an spatial frame canonically isomorphic to the topology of Max(M). We characterize the soberness of Max(M) in terms of the point space of that frame. Beside of this, we study the regularity of an spatial frame associated to M given by annihilator conditions.