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Spring 2020 Kronecker function rings and power series rings
Gyu Whan Chang
J. Commut. Algebra 12(1): 27-51 (Spring 2020). DOI: 10.1216/jca.2020.12.27

## Abstract

Let $D$ be an integral domain with quotient field $K$, $X$ be an indeterminate over $D$, $D\left[\phantom{\rule{-0.17em}{0ex}}\left[X\right]\phantom{\rule{-0.17em}{0ex}}\right]$ be the power series ring over $D$, and $c\left(f\right)$ be the ideal of $D$ generated by the coefficients of $f\in D\left[\phantom{\rule{-0.17em}{0ex}}\left[X\right]\phantom{\rule{-0.17em}{0ex}}\right]$. We will say that a star operation $\ast$ on $D$ is a c-star operation if (i) $c{\left(fg\right)}^{\ast }={\left(c\left(f\right)c\left(g\right)\right)}^{\ast }$ for all $0\ne f,g\in D\left[\phantom{\rule{-0.17em}{0ex}}\left[X\right]\phantom{\rule{-0.17em}{0ex}}\right]$ and (ii) ${\left(AB\right)}^{\ast }\subseteq {\left(AC\right)}^{\ast }$ implies ${B}^{\ast }\subseteq {C}^{\ast }$ for all nonzero fractional ideals $A,B,C$ of $D$. Assume that $D$ admits a c-star operation $\ast$, and let . Among other things, we show that $Kr\left(\phantom{\rule{-0.17em}{0ex}}\left(D,\ast \right)\phantom{\rule{-0.17em}{0ex}}\right)$ is a Bézout domain, $D$ is completely integrally closed, the $v$-operation on $D$ is a c-star operation, and $Kr\left(\phantom{\rule{-0.17em}{0ex}}\left(D,v\right)\phantom{\rule{-0.17em}{0ex}}\right)$ is a completely integrally closed Bézout domain. We also show that if $V$ is a rank-one valuation domain, then the $v$-operation on $V$ is a c-star operation, $Kr\left(\phantom{\rule{-0.17em}{0ex}}\left(V,v\right)\phantom{\rule{-0.17em}{0ex}}\right)$ is a rank-one valuation domain, and $Kr\left(\phantom{\rule{-0.17em}{0ex}}\left(V,v\right)\phantom{\rule{-0.17em}{0ex}}\right)$ is a DVR if and only if $V$ is a DVR. Using this result, we show that if $D$ is a generalized Krull domain, then $Kr\left(\phantom{\rule{-0.17em}{0ex}}\left(D,v\right)\phantom{\rule{-0.17em}{0ex}}\right)$ is a one-dimensional generalized Krull domain.

## Citation

Gyu Whan Chang. "Kronecker function rings and power series rings." J. Commut. Algebra 12 (1) 27 - 51, Spring 2020. https://doi.org/10.1216/jca.2020.12.27

## Information

Received: 1 January 2017; Revised: 2 June 2017; Accepted: 18 June 2017; Published: Spring 2020
First available in Project Euclid: 13 May 2020

zbMATH: 07211323
MathSciNet: MR4097054
Digital Object Identifier: 10.1216/jca.2020.12.27

Subjects:
Primary: 13A15, 13F05, 13F25, 13F30  