Star operations on overrings of Noetherian domains (Q888849)

The paper under review is the continuation of the authors' earlier paper [\textit{E. Houston} and \textit{M. H. Park}, J. Algebra 407, 105--134 (2014; Zbl 1302.13004)]. Let \(R\) be an integral domain and \(\mathrm{Star}(R)\) the set of star operations on it. The goal of this paper is to study the relationship between \(|\mathrm{Star}(R)|\) and \(|\mathrm{Star}(T)|\) for each overring \(T\) of \(R\). Call a domain \(R\) star regular if \(|\mathrm{Star}(R)| \geq |\mathrm{Star}(T )|\) for each overring \(T\) of \(R\). It was proved, for a local Noetherian domain with infinnite residue field, that if \(1\leq |\mathrm{Star}(R)|<\infty\), then \(R\) is star regular. Conversely, the authors showed that if \(R\) is a nonlocal Noetherian domain with \(|\mathrm{Star}(T )|<\infty\) for each proper overring \(T\) of \(R\), then \(|\mathrm{Star}(R)|<\infty\). On the other hand, if \(R\) is a local Noetherian domain with \(|\mathrm{Star}(T )|<\infty\) for each proper overring \(T\) of \(R\), then \(\dim(R)=1\).