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Paper   IPM / M / 9532
School of Mathematics
  Title:   Zariski-like topology on the classical prime spectrum of a module
  Author(s):  M. Behboodi (Joint with M. J. Noori)
  Status:   Published
  Journal: Bull. Iranian Math. Soc.
  Vol.:  35
  Year:  2009
  Pages:   253-269
  Supported by:  IPM
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 abmP for a, bR, and mM, implies that amP or bmP. 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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