A GENERALIZATION OF THE ZARISKI TOPOLOGY OF ARBITRARY RINGS FORMODULES

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) : Annl(M/P) = P}. If

SpecP (R M ) ̸= ∅, then PP :=
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 l.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 con- ditions on l.Spec(RM) (with the Zariski topology). Also, we topologies l.Spec(R M ) with the patch topology, and show that for every Noetherian left R-module M, l.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 l.Spec(RM) with the Zariski topology is a spectral space, i.e., l.Spec(RM) is homeomorphic to Spec(S) for some commuta- tive ring S. Also, as an application we show that for any ring R with ACC on ideals Spec(R) is a spectral space.