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Power Series Rings Over Prüfer $v$-multiplication Domains, II

  Published:2016-08-17
 Printed: Mar 2017
  • Gyu Whan Chang,
    Department of Mathematics Education, Incheon National University, Incheon 22012, Korea
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Abstract

Let $D$ be an integral domain, $X^1(D)$ be the set of height-one 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 height-one maximal $t$-ideal $M$ of $D$.
Keywords: Krull domain, P$v$MD, multiplicatively closed set of ideals, power series ring Krull domain, P$v$MD, multiplicatively closed set of ideals, power series ring
MSC Classifications: 13A15, 13F05, 13F25 show english descriptions Ideals; multiplicative ideal theory
Dedekind, Prufer, Krull and Mori rings and their generalizations
Formal power series rings [See also 13J05]
13A15 - Ideals; multiplicative ideal theory
13F05 - Dedekind, Prufer, Krull and Mori rings and their generalizations
13F25 - Formal power series rings [See also 13J05]
 

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