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In order to prove a statement, I need to prove the following claim :

If $R $ is an (not necessarily commutative) Artinian ring and $M $ is a finitely generated $R $-module, then $M$ is Noetherian.

I know that If $R $ is an Artinian ring and $M $ is a finitely generated $R $-module, then $M$ is Artinian.

What about the claim?Is there any hint to prove that? Or any counterexample?

Thank you

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  • $\begingroup$ Is your ring $R$ commutative? $\endgroup$ – Alex Wertheim Dec 9 '16 at 7:58
  • $\begingroup$ @AlexWertheim no $\endgroup$ – user115608 Dec 9 '16 at 8:04
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    $\begingroup$ Are you aware of the Hopkins-Levitzki theorem? $\endgroup$ – Alex Wertheim Dec 9 '16 at 8:11
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    $\begingroup$ If the ring is Artinian then it is also Noetherian. Finitely generated modules over a Noetherian ring are Noetherian. $\endgroup$ – Levent Dec 9 '16 at 8:18
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    $\begingroup$ @user115608 the Hopkins Levitzki theorem. Alex already mentioned it. You should have looked it up. $\endgroup$ – rschwieb Dec 9 '16 at 12:20
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By the Hopkins-Levitzki theorem, it follows that an artinian ring is noetherian.

So you have that as your ring is artinian, then it is noetherian. And any finitely generated module over a noetherian ring is noetherian.

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