Intersections of quotient rings and Pruefer v-multiplication domains

Abstract

Let $D$ be an integral domain with quotient field $K$. Call an overring $S$ of $D$ a subring of $K$ containing $D$ as a subring. A family $\{S_\lambda\mid\lambda \in \Lambda \}$ of overrings of $D$ is called a defining family of $D$, if $D = \bigcap\{S_\lambda\mid\lambda \in \Lambda \}$. Call an overring $S$ a sublocalization of $D$, if $S$ has a defining family consisting of rings of fractions of $D$. Sublocalizations and their intersections exhibit interesting examples of semistar or star operations. We show as a consequence of our work that domains that are locally finite intersections of Pr\"ufer $v$-multiplication (respectively, Mori) sublocalizations turn out to be Pr\"ufer $v$-multiplication domains (respectively, Mori); in particular, for the Mori domain case, we reobtain a special case of \cite[Th\'eor\`eme 1]{Q} and \cite[Proposition 3.2]{de}. We also show that, more than the finite character of the defining family, it is the finite character of the star operation induced by the defining family that causes the interesting results. As a particular case of this theory, we provide a purely algebraic approach for characterizing Pr\"ufer $v$-multiplication domains as a subclass of the class of essential domains (see also \cite[Theorem 2.4]{FT}).