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| Paper IPM / M / 11185 |
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| Abstract: | |
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Given a stable semistar operation of finite type ∗ on
an integral domain D, we show that it is possible to define in a
canonical way a stable semistar operation of finite type ∗[X]
on the polynomial ring D[X], such that, if n:=∗-dim(D),
then n+1 ≤ ∗[X]-dim(D[X]) ≤ 2n+1. We also
establish that if D is a ∗-Noetherian domain or is a
Prüfer ∗-multiplication domain, then
∗[X]-dim(D[X])=∗-dim(D)+1. Moreover we
define the semistar valuative dimension of the domain D, denoted
by ∗-dimv(D), to be the maximal rank of the
∗-valuation overrings of D. We show that
∗-dimv(D)=n if and only if
∗[X1,…,Xn]-dimv(D[X1,…,Xn])=2n, and that if
∗-dimv(D) < ∞ then
∗[X]-dimv(D[X])=∗-dimv(D)+1. In general
∗-dim(D) ≤ ∗-dimv(D) and equality holds if D is
a ∗-Noetherian domain or is a Prüfer
∗-multiplication domain. We define the ∗-Jaffard domains
as domains D such that ∗-dim(D) < ∞ and
∗-dim(D)=∗-dimv(D). As an application,
∗-quasi-Prüfer domains are characterized as domains D
such that each (∗,∗′)-linked overring T of D, is a
∗′-Jaffard domain, where ∗′ is a stable semistar
operation of finite type on T. As a consequence of this result we
obtain that a Krull domain D, must be a wD-Jaffard domain.
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