Let $S$ be a subring of a commutative Noetherian ring $R$. Then $R$ is finitely generated as an $S$-module?

Question

Let $S$ be a subring of a commutative Noetherian ring $R$. Then how can I show that $R$ is finitely generated as an $S$-module?

Answer 1

They assume that $R$ is finitely generated over $S$. In general this is not true. An example where this is not the case would be $R=\mathbb{R}[x]$ and $S=\mathbb{R}$.

Answer 2

It is not possible to show. For example, $\Bbb Q$ is a subring of the Noetherian ring $\Bbb R$ but of course $\Bbb R_\Bbb Q$ is not finitely generated.