A GENERALIZATION OF THE ZARISKI TOPOLOGY OF ARBITRARY RINGS FOR MODULES

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∈ Spec(R), we define SpecP (RM )= {P ∈ Spec(RM ): Ann� (M/P )= P} .I f SpecP (RM ) � ∅ ,t henPP := P ∈SpecP(R M) P is a prime submodule of M and P ∈ SpecP (RM ). A prime submodule Q of M is called a lower prime submodule provided Q = PP for some P∈ 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.