There is a proposition:
Let $A$ be a Noetherian integral domain, $B$ be an integral domain and $A \subset B$ a finite extension.
Then $A\subset B\cap Q(A)$ is also a finite extension where $Q(A)$ is field of fractions of $A$.
The proof is easy. Since $B$ is f.g $A$-module and $A$ is a Noetherian ring, $B$ is a Noetherian $A$-module. Hence $B\cap Q(A)$, which is submodule of $B$ is also finite.
Do you have any counterexample where $A$ is not Noetherian?
