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 $R$ ?)
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