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Let $A$ be a commutative Noetherian ring with identity and $M,N$ two finitely generated $A$-modules.

How to show $\operatorname{Hom}_A(M,N)$ is a finitely generated $A$-module?

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Consider the surjection $A^n \to M$. This induces an injection $Hom_A(M,N) \to Hom_A(A^n, N) = N^n$. So this is a submodule of a finitely generated module and by the Noetherian condition, we are done.

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