If $A\subseteq B$ where $A,B$ are commutative domains and $B$ is a finitely generated $A$ module, is $\operatorname{Frac}(A)\subseteq \operatorname{Frac}(B)$ a finite field extension?
I know this extension is algebraic and every element of Frac$(B)$ satisfies a monic polynomial with coefficients in $A$. I am not sure how to prove this extension is finite and not aware of any counterexamples.