Not sure how to show a module homomorphism to a finitely generated module is an isomorphism..

Asked Sep 15 '23 at 14:08

Active Sep 17 '23 at 05:30

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I have the question: Let $M$, $N$ be modules over a ring $R$ with homomorphisms $f, g : N \longrightarrow M$ such that $f$ is surjective and $g$ is injective. Show that: (1) $f$ is an isomorphism if $M$ is finitely generated.

No clue how to go about this, also not sure how/if $g$ is relevant at all.

*Note, I believe this is different from Orzech's theorem as in this case, N is not a submodule of M.

Thank you for any help.

edited Sep 17 '23 at 05:30

asked Sep 15 '23 at 14:08

mad_scientist

Since you gave the tag [tag:commutative-algebra] I assume everything here is commutative? – Trebor Sep 15 '23 at 14:20
Hint: think to the case of vector spaces – Romulus Augustulus Sep 15 '23 at 16:09
@Trebor yes R is commutative. – mad_scientist Sep 16 '23 at 03:46
@mad_scientist *Note, I believe that one can replace $N$ by $g(N)$. – user26857 Sep 17 '23 at 10:35

Not sure how to show a module homomorphism to a finitely generated module is an isomorphism..

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