Semistar-Krull and valuative dimension of integral domains

چکیده مقاله

Given a stable semistar operation of finite type $\star$ on
an integral domain $D$, we show that it is possible to define in a
canonical way a stable semistar operation of finite type $\star[X]$
on the polynomial ring $D[X]$, such that, if $n:=\star$-$\dim(D)$,
then $n+1\leq \star[X]\text{-}\dim(D[X])\leq 2n+1$. We also
establish that if $D$ is a $\star$-Noetherian domain or is a
Pr\"{u}fer $\star$-multiplication domain, then
$\star[X]\text{-}\dim(D[X])=\star\text{-}\dim(D)+1$. Moreover we
define the semistar valuative dimension of the domain $D$, denoted
by $\star$-$\dim_v(D)$, to be the maximal rank of the
$\star$-valuation overrings of $D$. We show that
$\star$-$\dim_v(D)=n$ if and only if
$\star[X_1,\cdots,X_n]$-$\dim(D[X_1,\cdots,X_n])=2n$, and that if
$\star$-$\dim_v(D)<\infty$ then
$\star[X]$-$\dim_v(D[X])=\star$-$\dim_v(D)+1$. In general
$\star$-$\dim(D)\leq\star$-$\dim_v(D)$ and equality holds if $D$ is
a $\star$-Noetherian domain or is a Pr\"{u}fer
$\star$-multiplication domain. We define the $\star$-Jaffard domains
as domains $D$ such that $\star$-$\dim(D)<\infty$ and
$\star$-$\dim(D)=\star$-$\dim_v(D)$. As an application,
$\star$-quasi-Pr\"{u}fer domains are characterized as domains $D$
such that each $(\star,\star')$-linked overring $T$ of $D$, is a
$\star'$-Jaffard domain, where $\star'$ 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 $w_D$-Jaffard domain.