We know that Finitely generated modules over a Noetherian ring are Noetherian and finitely generated modules over a Artinian ring are artinian.
Now I want to know whether the converse is also right, that is: Are noetherian modules over a noetherian ring finitely generated? Are artinian modules over a artinian ring finitely generated?
To noetherian case, I think it is right: Let $M$ be a left noetherian module over a noetherian ring $R$. Suppose $M$ is not finitely generated, take $m_1 \in M$, then $M_1=Rm_1 \subset M$. Since $M$ is not finitely generated, there is $m_2 \in M$ \ $M_1$,$M_2 :=M_1 +Rm_2$, then $M_1 \subset M_2 $. Take $m_3 \in M$\ $M_2$, $M_3 := M_2 +Rm_3$, then $M_1 \subset M_2 \subset M_3$. Continue this process, we get an infinite increasing chain of submodules of $M$. This contradicts with $M$ noetherian. So $M$ is finitely generated.
I can't make sure the above is right, thank you for checking. Also I don't know whether the artinian case is right or wrong. Thank you for any help.