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# 12 - M. Behboodi (with S. H. Shojaee), A generalization of the Zariski topology of arbitrary rings for modules, East-West J. Math. 11(2), (2009), 165-183

Let M be a left R-module. The set of all prime submodules of M is called the spectrum of M and denoted by Spec(RM), and that of all prime ideals of R is denoted by Spec(R). For each P 2 Spec(R), we define SpecP(RM) = {P 2 Spec(RM) : Ann`(M/P) = P}. If SpecP(RM) 6= ;, then PP := TP2SpecP(RM) P is a prime submodule of M and P 2 SpecP(RM). A prime submodule Q of M is called a lower prime submodule provided Q = PP for some P 2 Spec(R). We write `.Spec(RM) for the set of all lower prime submodules of M and call it lower spectrum of M. In this article, we study the relationships among various module-theoretic properties of M and the topological conditions on `.Spec(RM) (with the Zariski topology). Also, we topologies Spec(RM) with the patch topology, and show that for every Noetherian left R-module M, `.Spec(RM) with the patch topology is a compact, Hausdorff, totally disconnected space. Finally, by applying Hochster’s characterization of a spectral space, we show that if M is a Noetherian left R-module, then `.Spec(RM) with the Zariski topology is a spectral space, i.e., `.Spec(RM) is homeomorphic to Spec(S) for some commutative ring S. Also, as an application we show that for any ring R with ACC on ideals, Spec(R) is a spectral space.