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| Paper IPM / M / 8465 |
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| Abstract: | |
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Let M be a left R-module. The spectrum of M (denoted by
Spec(RM)) is the collection of all prime submodules of M and
the spectrum of R (denoted by Spec(R)) is the set of all
prime ideals of R. For each P ∈ Spec(R), we define
SpecP(RM)={P ∈ Spec(RM) :Annl(M/P)=P}. If SpecP(RM) ≠ ∅, then PP:=∩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 l.Spec(RM) for the
set of all lower prime submodules of M and call it lower
spectrum of M (clearly for any ring R, we have
l.Spec(RR)=Spec(R)). In this article, we study the
relationships among various module-theoretic properties of M and
the topological conditions on l.Spec(RM) (with the Zariski
topology). Also, we topologies l.Spec(RM) 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 commutative ring S. Also,
as an application we
show that for any ring R with ACC on ideals Spec(R) is a spectral space.
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