Power Series Rings Over Prüfer v-multiplication Domains. II

Abstract

Let $D$ be an integral domain, ${{X}^{1}}\left( D \right)$ be the set of height-one prime ideals of $D$, $\left\{ {{X}_{\beta }} \right\}$ and $\left\{ {{X}_{\alpha }} \right\}$ be two disjoint nonempty sets of indeterminates over $D$, $D\left[ \left\{ {{X}_{\beta }} \right\} \right]$ be the polynomial ring over $D$, and $D\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1}}$ be the first type power series ring over $D\left[ \left\{ {{X}_{\beta }} \right\} \right]$. Assume that $D$ is a Prüfer $v$-multiplication domain $\left( \text{P}v\text{MD} \right)$ in which each proper integral $t$-ideal has only finitely many minimal prime ideals (e.g., $t$-$\text{SFT}$$\text{P}v\text{MDs}$, valuation domains, rings of Krull type). Among other things, we show that if ${{X}^{1}}\left( D \right)\,=\,\phi$ or ${{D}_{p}}$ is a $\text{DVR}$ for all $P\,\in \,{{X}^{1}}\left( D \right)$, then $D\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1D-\left\{ 0 \right\}}}$ is a Krull domain. We also prove that if $D$ is a $t$-$\text{SFT}\text{P}v\text{MD}$, then the complete integral closure of $D$ is a Krull domain and $\text{ht}\left( M\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1}} \right)\,=\,1$ for every height-one maximal $t$-ideal $M$ of $D$.