I've read Proposition 4 and Corollary 5 in this paper.
Proposition 4 Let $R$ be a ring, $ S \subseteq R$ a multiplicative set and $M$ an $S$-finite $R$-module. Then $M$ is $S$ Noetherian if and only if the submodule of the form $PM$ are $S$-finite for each prime ideal $P$ of $R$ (disjoint from $S$).
Corollary 5 Let $R$ be a ring and $S\subseteq R$ a multiplicative set. Then $R$ is $S$-Noetherian if and only if every prime ideal of $R$ (disjoint from $S$) is $S$-finite.
I've proved Proposition 4 and I want to prove Corollary 5. I can't do that because in Proposition 4, $M$ is $S$-finite, but in Corollary 5, it didn't say anything about $S$-finiteness of $R$. Is every commutative ring could be $S$-finite ?