Power series rings over Pr\"ufer $v$-multiplication domains
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초록
Let $D$ be an integral domain, $\{X_{\alpha}\}$ be a nonempty set of indeterminates over $D$, and ${D[\![\{X_{\alpha}\}]\!]_1}$ be the first type power series ring over $D$. We show that if $D$ is a $t$-SFT Pr\"ufer $v$-multiplication domain, then ${D[\![\{X_{\alpha}\}]\!]_1}_{D - \{0\}}$ is a Krull domain, and $D[\![\{X_{\alpha}\}]\!]_1$ is a Pr\"ufer $v$-multiplication domain if and only if $D$ is a Krull domain.
Abstract
Let $D$ be an integral domain, $\{X_{\alpha}\}$ be a nonempty set of indeterminates over $D$, and ${D[\![\{X_{\alpha}\}]\!]_1}$ be the first type power series ring over $D$. We show that if $D$ is a $t$-SFT Pr\"ufer $v$-multiplication domain, then ${D[\![\{X_{\alpha}\}]\!]_1}_{D - \{0\}}$ is a Krull domain, and $D[\![\{X_{\alpha}\}]\!]_1$ is a Pr\"ufer $v$-multiplication domain if and only if $D$ is a Krull domain.
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