  Call a domain $R$ an sQQR-domain if each simple overring of $R$, i.e., each
ring of the form $R[u]$ with $u$ in the quotient field of $R$, is an
intersection of localizations of $R$. We characterize Pr\"ufer domains as
integrally closed sQQR-domains. In the presence of certain finiteness
conditions, we show that the sQQR-property is very strong; for instance, a Mori
sQQR-domain must be a Dedekind domain. We also show how to construct
sQQR-domains which have (non-simple) overrings which are not intersections of
localizations.
