$\hom_R(A,B)$ is finitely generated if $R$ is noetherian

Question

This is part of an exercise I'm doing, from Rotman Introduction to homological algebra.

Let $R$ be a commutative ring, and let $A$ and $B$ be finitely generated $R$-modules.

Then if $R$ is noetherian, prove that $\hom_R(A, B)$ is a finitely generated $R$-module.

I'm stuck on it, any hint ?

Answer 1

HINT: If $A$ is finitely generated, there is an epimorphism of modules of the form $$R^n\rightarrow A$$ for some $n$. This induce a monomorphism $$\text{hom}_R(A,B)\rightarrow \text{hom}_R(R^n,B)\text{.}$$ What can you say about $\text{hom}_R(R^n,B)$?