A new characterization of Pr\"ufer v-multiplication domains

Abstract

Let $D$ be an integral domain and $w$ be the so-called $w$-operation on $D$. In this note, we introduce the notion of $\ast (w)$-domains: $D$ is a $\ast (w)$-domain if $((\cap (x_i))(\cap (y_j)){)}_w=\cap (x_i{y}_j)$ for all nonzero elements $x_1,\dots ,x_m;y_1,\dots ,y_n$ of $D$. We then show that $D$ is a Pr\"ufer $v$-multiplication domain if and only if $D$ is a $\ast (w)$-domain and $A^{-1}$ is of finite type for all nonzero finitely generated fractional ideals $A$ of $D$.