Star operations on Prüfer v -multiplication domains

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 $|S_w(D)| \lt \infty$.