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| Paper IPM / M / 9532 |
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| Abstract: | |
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Let R be a commutative ring with identity and let M
be an R-module. A proper submodule P of M is called a
classical prime submodule if abm ∈ P for a, b ∈ R, and
m ∈ M, implies that am ∈ P or bm ∈ P. The classical
prime spectrum Cl.Spec(M) is defined to be the set of all
classical prime submodules of M. The aim of this paper is to
introduce and study a topology on Cl.Spec(M), which generalize
the Zariski topology of R to M, called Zariski-like topology
of M. In particular, we investigate this topological space from
the point of view of spectral spaces. It is shown that if M is
a Noetherian (or an Artinian) R-module, then Cl.Spec(M) with
the Zariski-like topology is a spectral space, i.e., there exists
a commutative ring S such that Cl.Spec(M) with the
Zariski-like topology is homeomorphic to Spec(S) with the usual
Zariski topology.
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