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Paper   IPM / M / 9551
School of Mathematics
  Title:   On semistar Nagata rings, Prufer-like domains and semistar going-down domains
  Author(s):  P. Sahandi (Joint with D. E. Dobbs)
  Status:   Published
  Journal: Houston J. Math.
  Vol.:  37
  Year:  2011
  Pages:   715-731
  Supported by:  IPM
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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