Zariski-like topology on the classical prime spectrum of a modules

Abstract. Let R be a commutative ring with identity and let M be an R-module. A proper submodule P of M is called a classical prime submodule if abm ∈ P for a, b ∈ R, and m ∈ M, implies that am ∈ P or bm ∈ P. The classical prime spectrum Cl.Spec(M) is defined to be the set of all classical prime submodules of M. The aim of this paper is to introduce and study a topology on Cl.Spec(M), which generalizes the Zariski topology of R to M, called Zariski-like topology of M. In particular, we investigate this topological space from the point of view of spectral spaces. It is shown that if M is a Noetherian (or an Artinian) R-module, then Cl.Spec(M) with the Zariski-like topology is a spectral space, i.e., there exists a commutative ring S such that Cl.Spec(M) with the Zariski-like topology is homeomorphic to Spec(S) with the usual Zariski topology. 1