I found in this discussion that the fact that a
surjective endomorphism $e: M \to M$ of a finitely generated $R$-module ($R$ Noetherian) is an isomorphism
can be showed by so called 'Noetherian overkill'. Why in this situationthe explotation of the Noetherian assumption is an 'overkill' argument?
Note: I know some proofs (like this one: Surjective endomorphisms of finitely generated modules are isomorphisms ; see eg in Georges' answer a nice application of lemma of Nakayama) of this claim but in
this question I just want to find out why noetherianess here is in certain sense 'over the top'.
Best Answer
Because, as the commenter said, the same thing can be proven without the Noetherian hypothesis. Therefore the assumption is "overkill" meaning "unnecessary."
There is not a special technique called "noetherian overkill." From the context you are reading, it just refers to results which are often stated with the Noetherian hypothesis, which do not actually require it.