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Paper   IPM / M / 8849
School of Mathematics
  Title:   Going-Down and semistar operations
  Author(s):  P. Sahandi (Joint with D. E. Dobbs)
  Status:   Published
  Journal: J. Algebra Appl.
  Vol.:  8
  Year:  2009
  Pages:   83-104
  Supported by:  IPM
If DT is an extension of (commutative integral) domains and ∗ (resp., ∗′) is a semistar operation on D (resp., T), we define what it means for DT to satisfy the (∗,∗′)−GD property. Sufficient conditions are given for (∗,∗′)−GD, generalizing classical sufficient conditions for GD such as flatness, openness of the contraction map of spectra and the hypotheses of the classical going-down theorem. If ∗ is a semistar operation on a domain D, we define what it means for D to be a ∗-GD domain, generalizing the notion of a going-down domain. In determining whether a domain D is a ~∗-GD domain, the domain extensions T of D for which (~∗,∗′)−GD is tested can be the ~∗-valuation overrings of D, the simple overrings of D, or all T. PMDs are characterized as the ~∗-treed (resp., ~∗-GD) domains D which are ~∗-finite conductor domains such that D~ is integrally closed. Several characterizations are given of the ~∗-Noetherian domains D of ~∗-dimension 1 in terms of the behavior of the (∗,∗′)-linked overrings of D and the ∗-Nagata rings Na(D,∗).

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