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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?

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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.

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