A class of integral domains whose integral closures are small submodules of the quotient field

Abstract

Let $(V,M)$ be a valuation domain which is distinct from its quotient field $K$, and let $\pi :V\to V/M$ be the canonical surjection. Let $D$ be a subring of $V/M$. It is proved that the pullback $R:={\pi }^{-1}(D)$ has the property that $V$ (and hence each integral overring of $R$) is a small $R$-submodule of $K$. Applications include all classical $D+M$ constructions and locally pseudo-valuation domains.