Extension of Integral Domains 
Let $S\subset R$ be an extension of integral domains. If the ideal $(S:R)=\{s\in S\mid sR\subseteq S\}$ is finitely generated, show that $R$ is integral over $S$.

My first attempt was to show that $R$ is finitely generated as an $S$-module, then the extension is immediately integral. Is this always the case though? I am having some trouble figuring out how to show this. Also, I don't see where the fact that they are integral domains could come into play.

Hint. The claim follows from the standard determinant trick.
