Let be a DVR and be a free -module of rank .

If is a surjective -linear map then is an isomorphism.

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- Apr 11th 2011, 11:15 AMKaKaAbout some module over a DVR
Let be a DVR and be a free -module of rank .

If is a surjective -linear map then is an isomorphism. - Apr 13th 2011, 01:48 AMNonCommAlg
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.