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Let $A$ be a Noetherian ring and $M$ be a $A$-module.

Then if I am correct, $M$ is Noetherian $A$-module if $M$ is finitely generated $A$-module.

Does the converse hold true ?

i.e., Is a Noetherian module always finitely generated over Noetherian ring ?

It seems to me that Noetherian module is stronger than finitely generated module, because every submodule (and hence itself) of Noethrian module is finitely generated.

Is a Noetherian module over any ring a finitely generated module ?

Thanks

MAS
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  • Why negative vote ? You can instead suggest to improve the question – MAS Oct 19 '22 at 15:56
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    Every ring is finitely generated over itself but it may not be necessarily Noetherian.For eg: take polynomial ring over infinite variables it is not noetherian but it is f.g. module over itself. – Dgarg12 Oct 20 '22 at 07:28

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