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| Paper IPM / M / 8849 |
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| Abstract: | |
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If D ⊆ T is an extension of (commutative integral) domains
and ∗ (resp., ∗′) is a semistar operation on D
(resp., T), we define what it means for D ⊆ T 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. P∗MDs 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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