I saw this lemma in some lecture notes, there was no proof given nor a reference, only a statement that it can be found in any text-book on commutative algebra. I checked several but couldn't find it.
Let $F$ be a field, $B$ an $F$-algebra which is an integral domain, $A$ a Noetherian subalgebra of $B$, and such that $B$ is integral over $A$, and the field of fractions of $B$ is a finite extension of the field of fractions of $A$. Then $B$ is a finitely generated $F$-algebra if and only if $A$ is a finitely generated $F$-algebra. (I'm mainly interested in the 'if' part).
Has anyone seen this before, knows a proof or where to find one? Thanks.