### 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 \in Spec(R),

we define SpecP(RM)={P \in Spec(RM):Ann\ell (M/P)=P}. If Spec_P(RM) \neq \empty, then PP :=\ell P?SpecP(RM)Pis a prime submodule

ofMandP?SpecP(RM). A prime submodule QofMis called a

lower prime submodule provided Q=PPfor some P?Spec(R). We

write\ell .Spec(RM) for the set of all lower prime submodules of Mand

call itlower spectrum ofM. In this article, we study the relationships

among various module-theoretic properties ofMand the topological conditions on\ell .Spec(RM) (with the Zariski topology). Also, we topologies

\ell .Spec(RM) with the patch topology, and show that for every Noetherian

left R-module M, \ell .Spec(RM) with the patch topology is a compact,

Hausdorff, totally disconnected space. Finally, by applying Hochsters

characterization of a spectral space, we show that ifMis a Noetherian

left R-module, then \ell .Spec(RM) with the Zariski topology is a spectral

space, i.e.,\ell .Spec(RM) is homeomorphic to Spec(S) for some commutative ringS. Also, as an application we show that for any ring Rwith

ACC on ideals Spec(R) is a spectral space.

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 \in Spec(R),

we define SpecP(RM)={P \in Spec(RM):Ann\ell (M/P)=P}. If Spec_P(RM) \neq \empty, then PP :=\ell P?SpecP(RM)Pis a prime submodule

ofMandP?SpecP(RM). A prime submodule QofMis called a

lower prime submodule provided Q=PPfor some P?Spec(R). We

write\ell .Spec(RM) for the set of all lower prime submodules of Mand

call itlower spectrum ofM. In this article, we study the relationships

among various module-theoretic properties ofMand the topological conditions on\ell .Spec(RM) (with the Zariski topology). Also, we topologies

\ell .Spec(RM) with the patch topology, and show that for every Noetherian

left R-module M, \ell .Spec(RM) with the patch topology is a compact,

Hausdorff, totally disconnected space. Finally, by applying Hochsters

characterization of a spectral space, we show that ifMis a Noetherian

left R-module, then \ell .Spec(RM) with the Zariski topology is a spectral

space, i.e.,\ell .Spec(RM) is homeomorphic to Spec(S) for some commutative ringS. Also, as an application we show that for any ring Rwith

ACC on ideals Spec(R) is a spectral space.

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