Kronecker function rings and power series rings

Abstract

Let $D$ be an integral domain with quotient field $K$, $X$ be an indeterminate over $D$, $D\left[\left[X\right]\right]$ be the power series ring over $D$, and $c\left(f\right)$ be the ideal of $D$ generated by the coefficients of $f\in D\left[\left[X\right]\right]$. 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 $0\ne f,g\in D\left[\left[X\right]\right]$ 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 $Kr\left(\left(D,\ast \right)\right)=\left\{f∕g\mid f,g\in D\left[\left[X\right]\right],g\ne 0,\text{and }c\left(f\right)\subseteq c{\left(g\right)}^{\ast }\right\}$. Among other things, we show that $Kr\left(\left(D,\ast \right)\right)$ is a Bézout domain, $D$ is completely integrally closed, the $v$-operation on $D$ is a c-star operation, and $Kr\left(\left(D,v\right)\right)$ 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, $Kr\left(\left(V,v\right)\right)$ is a rank-one valuation domain, and $Kr\left(\left(V,v\right)\right)$ 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 $Kr\left(\left(D,v\right)\right)$ is a one-dimensional generalized Krull domain.