Abstract view
Power Series Rings Over Prüfer $v$multiplication Domains, II


Published:20160817
Printed: Mar 2017
Gyu Whan Chang,
Department of Mathematics Education, Incheon National University, Incheon 22012, Korea
Abstract
Let $D$ be an integral domain, $X^1(D)$ be the set of heightone
prime ideals of $D$,
$\{X_{\beta}\}$ and $\{X_{\alpha}\}$ be
two disjoint nonempty sets of indeterminates over $D$,
$D[\{X_{\beta}\}]$ be the polynomial ring over $D$, and
$D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1$ be the first type
power series ring over $D[\{X_{\beta}\}]$.
Assume that $D$ is a Prüfer $v$multiplication domain (P$v$MD)
in which each proper integral $t$ideal has only finitely many
minimal prime ideals
(e.g., $t$SFT P$v$MDs, valuation domains, rings of Krull type).
Among other things, we show that if
$X^1(D) = \emptyset$ or $D_P$ is a DVR for all $P \in X^1(D)$,
then
${D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1}_{D  \{0\}}$ is a
Krull domain.
We also prove that if $D$ is a $t$SFT P$v$MD, then the complete
integral closure of $D$ is a Krull domain and
ht$(M[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1)$ = $1$ for every
heightone maximal $t$ideal $M$ of $D$.