Zariski-like topology on the classical prime spectrum of a module (Q841431)

The authors offer a generalization of the Zariski topology of rings to modules by defining a topology on the set of all classical prime submodules of an \(R\)-module \(M\), denoted by \(\text{Cl.Spec}(M)\). They call this type of topology a Zariski-like topology of \(M\) and they investigate it from the point of view of spectral spaces. After giving several results on classical top modules (cf. Definition 1.1) and top-modules, they are able to prove the equivalence of the notions of classical top-module, top-module and multiplication module for a finitely generated \(R\)-module \(M\) (cf. Theorem 2.7). They also, among others, prove that \(\text{Spec(M) = Cl.Spec(M)}\) for an Artinian module \(M\) (cf. Proposition 2.8). Furthermore, they prove the important result that if \(M\) is a Noetherian (or an Artinian) \(R\)-module, then \(\text{Cl.Spec(M}\) with Zariski-like topology is a spectral space (cf. Theorem 3.10 and Theorem 3.11). They dedicate the last section of the paper to provide a correct version of [\textit{Ch.-P. Lu}, Houston J. Math. 25, No. 3, 417--432 (1999; Zbl 0979.13005), Proposition 5.2(3) and Proposition 6.3].