Journal of Commutative Algebra

Star operations on Prüfer v -multiplication domains

Gyu Whan Chang

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Abstract

Let $D$ be an integrally closed domain, $S(D)$ the set of star operations on $D$, $w$ the $w$-operation, and $S_w(D) = \{* \in S(D) \mid w \leq *\}$. Let $X$ be an indeterminate over $D$ and $N_v = \{f \in D[X] \mid c(f)_v = D\}$. In this paper, we show that, if $D$ is a Pr\"ufer $v$-multiplication domain (P$v$MD), then $|S_w(D)| = |S_w(D[X])| = |S(D[X]_{N_v})|$. We prove that $D$ is a P$v$MD if and only if $|\{* \in S_w(D) \mid *$ is of finite type$\}|\lt \infty$. We then use these results to give a complete characterization of integrally closed domains $D$ with $|S_w(D)| \lt \infty$.

Article information

Source
J. Commut. Algebra, Volume 7, Number 4 (2015), 523-543.

Dates
First available in Project Euclid: 19 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.jca/1453211672

Digital Object Identifier
doi:10.1216/JCA-2015-7-4-523

Mathematical Reviews number (MathSciNet)
MR3451354

Zentralblatt MATH identifier
1329.13003

Subjects
Primary: 13A15: Ideals; multiplicative ideal theory 13G05: Integral domains

Keywords
Star operation $w$-operation P$v$MD $D[X]_{N_v}$ integrally closed domain

Citation

Chang, Gyu Whan. Star operations on Prüfer v -multiplication domains. J. Commut. Algebra 7 (2015), no. 4, 523--543. doi:10.1216/JCA-2015-7-4-523. https://projecteuclid.org/euclid.jca/1453211672


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