I am given that $R$ is a Noetherian ring and $S \subseteq R[x]$ is a ring. How can I prove that if $R[x]$ is a finitely generated $S$-module then $S$ is a finitely generated $R$-algebra.
I am thinking of using Artin -Tate lemma and induction but not sure of how to write the proof for the induction part. Could someone please walk me through the proof. Any help is appreciated!