Let $A \subset B \subset C$ be subrings. If $C$ is finitely generated as an $A$-module, is $B$ finitely generated as an $A$-module?
$A\subseteq B\subseteq C$ ring extensions, $A\subseteq C$ finite/finitely-generated $\Rightarrow$ $A\subseteq B$ finite/finitely-generated?