I am trying to prove:
If $B$ is integral closure of $A$ in some field extnesion $L$ of the quotient field of $A$, then $S^{-1}B$ is the integral closure of $S^{-1}A$ in $L$.
I could prove:
(a) Let $A$ be a subring of a ring $B$, integral over $A$. Let $S$ be a multiplicative subset of $A$. Then $S^{-1}B$ is integral over $S^{-1}A$. If $A$ is integrally closed, then $S^{-1}A$ is integrally closed.
(b) Integral closure of $A$ in some field extension $L$ is integrally closed.
I am thinking to complete the problem as follows:
$S^{-1}B$ is integrally closed and integral over $S^{-1}A$ hence $S^{-1}B$ is integral closure of $S^{-1}A$ in $L$.
Problem is: if $C$ is integrally closed $C \subseteq L$ for some field, then the integral closure of $C$ in $L$ is $C$ (?)