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Let $R$ be a commutative ring with unity and $I \subseteq R$ an ideal. Prove: if $R/I$ is a Noetherian ring and $I/I^2$ is a finitely-generated $R$-module, then $I^{n-1}/I^n$ is a Noetherian $R$-module.

How can I show this?

I am trying by induction. For n=2 is given as $I/I^2$ is f.g. $R/I$ module and $R/I$ is Noetherian ring. For n=3 we have to show $I^2/I^3$ is Noetherian $R$ module. So I have to construct a s.e.s. involving $I/I^2 $ and $I^2/I^3$. But I can't construct such a sequence. Help me. Thank you.

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1 Answer 1

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It suffices to show that $I^{n-1}/I^n$ is finitely generated as an $R$-module, because then it is also finitely generated, hence Noetherian (since $R/I$ is Noetherian), as an $R/I$-module, and finally being Noetherian over $R/I$ or over $R$ is the same.

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