# A GENERALIZATION OF THE ZARISKI TOPOLOGY OF ARBITRARY RINGS FOR MODULES

### Abstract

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}. If SpecP(RM) = ∅, thenPP := P∈SpecP(RM) 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.