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Paper IPM / M / 8465  


Abstract:  
Let M be a left Rmodule. The spectrum of M (denoted by
Spec(_{R}M)) 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
Spec_{P}(_{R}M)={P ∈ Spec(_{R}M) :Ann_{l}(M/P)=P}. If Spec_{P}(_{R}M) ≠ ∅, then P_{P}:=∩_{P ∈ SpecP(RM)}P is a prime submodule of M and P ∈
Spec_{P}(_{R}M). A prime submodule Q of M is called a
lower prime submodule provided Q=P_{P} for some
P ∈ Spec(R). We write l.Spec(_{R}M) 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(_{R}R)=Spec(R)). In this article, we study the
relationships among various moduletheoretic properties of M and
the topological conditions on l.Spec(_{R}M) (with the Zariski
topology). Also, we topologies l.Spec(_{R}M) with the patch
topology, and show that for every Noetherian left Rmodule M,
l.Spec(_{R}M) 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 Rmodule, then l.Spec(_{R}M) with
the Zariski topology is a spectral space, i.e., l.Spec(_{R}M)
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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