We know that it's possible subring of a commutative Noetherian ring is not Noetherian (for example: Subring of a finitely generated Noetherian ring need not be Noetherian?).
But if $S$ be a subring of commutative Noetherian ring $R$, and $R$ is finitely generated as a $S$-module, is it true that we conclude $S$ is a Noetherian ring?
Best Answer
The answer to your question is positive and moreover it's a celebrated theorem, called the Eakin-Nagata's Theorem. For a proof you can look, e.g., here.