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Paper   IPM / M / 8465
School of Mathematics
  Title:   A generalization of the Zariski topology of arbitarary rings for modules
  Author(s):  M. Behboodi (Joint with S. H. Shojaee)
  Status:   Published
  Journal: East-West J. Math.
  Vol.:  11
  Year:  2009
  Pages:   165-183
  Supported by:  IPM
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)={PSpec(RM) :Annl(M/P)=P}. If SpecP(RM) ≠ ∅, then PP:=∩PSpecP(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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