Finitely presented modules and homomorphism modules

Question

Suppose $M$ and $N$ are two finitely presented $R$-modules, where $R$ is a commutative ring. Suppose in addition that $M$ is a projective $R$-module. How does one show that $\mathrm{Hom}_R(M,N)$ is a finitely presented $R$-module?
I tried using the two exact sequences given by the presentations of $M$ and $N$ to build a presentation of $\mathrm{Hom}_R(M,N)$ but I did not find one.
Thanks in advance for your help!

Answer 1

Let $M$ be a finitely generated projective $R$-module, and let $N$ be a finitely presented $R$-module. Then $\mathrm{hom}_R(M,N)$ is finitely presented:

In fact, $M$ is a direct summand of some finite free $R$-module $R^n$. It follows that $\mathrm{hom}_R(M,N)$ is a direct summand of $\mathrm{hom}_R(R^n,N)$. This module is isomorphic to $N^n$, which is finitely presented.

We conclude with the lemma that direct summands of finitely presented modules are also finitely presented.