the problem is a very special case of this one: if is a ring (doesn't even have to be commutative) and is a (left or right) Noetherian -module, then every -linear surjective map is an isomorphism. the proof is very easy:

look at the chain of submodules

,which has to stop at some point because is Noetherian. so

,for some positive integer . now suppose that , for some . we have , for some , because is surjective. thus and hence . so . this proves that is injective, which is what we need.