Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $R$ ?)

share|cite|improve this question

1 Answer 1

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:

For the Akizuki-Hopkins-Levitzki Theorem check

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.