Let $K$ be a field. Interestingly, the integrally closed subrings of $K$ are characterized as the intersections of valuation rings.
What happens if we replace "valuation rings" with "local rings" above? Which subrings of a field occur as intersections of local rings containing them? That is, for a ring $R \subset K$, when do we have $R = \bigcap_{L \text{ local, } R \subset L \subset K } L$?
Every integral domain $R$ is the intersection of all of its localizations at maximal ideals (considered as subrings of its field of fractions $K$). Indeed, suppose $a\in K\setminus R$. Then $I=\{r\in R:ra\in R\}$ is a proper ideal of $R$, so it is contained in some maximal ideal $M$. We then see that $a\not\in R_M$, since every possible way to write $a$ as a fraction of elements of $R$ has denominator in $M$.