# Is any Finitely generated module over an Artinian ring , Noetherian?

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

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