Overring properties of $G$-domains

Abstract:

A commutative domain $R$ is called a strong $G$-domain if every overring between $R$ and the quotient field $K$ of $R$ is of the form $R[1/t]$ for some nonzero element $t$ of $R$. After characterizing valuation rings which are strong $G$-domains, the authors show that $R$ is a strong $G$-domain if and only if it is a finite intersection of valuation rings each of which is a strong $G$-domain. Using some results of R. W. Gilmer, Jr., the authors identify the strong $G$-domains in the class of all Prüfer domains. They reprove via Krull domains the theorem characterizing Noetherian $G$-domains, a result first proved by Artin and Tate. The authors also raise some relevant questions on related overring properties of $G$-domains.