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I have a left artinian ring $R$ and a finitely generated left $R$-module $M$, and a submodule $A$ of $M$. My question is : is $A$ necesseraly finitely generated ? (and is there a direct proof of this from the "artinianity" of $R$ ?)

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Any left Artinian ring is left Noetherian (Akizuki-Hopkins-Levitzki Theorem), and a finitely generated module over a left Noetherian ring is Noetherian.

Check under "Properties" in this wikipedia page: http://en.wikipedia.org/wiki/Noetherian_ring.

For the Akizuki-Hopkins-Levitzki Theorem check http://en.wikipedia.org/wiki/Hopkins%E2%80%93Levitzki_theorem

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Thanks. My original (homework) problem asked for a projective resolution of $M$ in finitely generated projectives (and the previous exercise was the existence of a composition series for $R$ ...). –  user52599 Dec 11 '12 at 11:46

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