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| Paper IPM / M / 9551 |
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| Abstract: | |
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Let ∗ be a semistar operation on a domain D. Then the
semistar Nagata ring \Na(D, ∗) is a treed domain
⇔ D is ~∗-treed and the
contraction map \Spec(\Na(D,∗))→\QSpec~∗(D)∪{0} is a bijection
⇔ D is a ~∗-treed and
~∗-quasi-Prüfer domain. Consequently, if D is a
~∗-Noetherian domain but not a field, then D is
~∗-treed if and only if
~∗-dim(D)=1. The ring \Na(D, ∗) is a
going-down domain if and only if D is a ~∗-\GD
domain and a ~∗-quasi-Prüfer domain. In general,
D is a P∗MD ⇔ \Na(D,∗) is an
integrally closed treed domain ⇔ \Na(D,∗) is
an integrally closed going-down domain. If P is a
quasi-∗-prime ideal of D, an induced stable semistar
operation of finite type, ∗/P, is defined on D/P. The
associated Nagata rings satisfy \Na(D/P,∗/P) ≅ \Na(D,∗)/P\Na(D,∗). If D is a P∗MD (resp., a
~∗-Noetherian domain; resp., a ∗-Dedekind
domain; resp., a ~∗-GD domain), then D/P is a
P(∗/P)MD (resp., a (∗/P)-Noetherian domain; resp., a
(∗/P)-Dedekind domain; resp., a (∗/P)-GD domain).
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