Finitely generated modules

Question

Suppose that $R$ is a commutative ring and $M$ and $N$ are finitely generated $R-$modules. What we can say about $Hom_R (M,N)$? is it a finitely generated $R$-module?

Answer 1

As said by a previous user of Mathematics Stack Exchange (user now deleted):

If $R$ is not noetherian it is not clear (to me) what kind of finiteness conditions on modules would imply that $\operatorname{Hom}_R(M,N)$ is finitely generated. Even for finitely presented modules this property fails. An example is the following: $R=K[X_1,\dots,X_n,\dots]/(X_1,\dots,X_n,\dots)^2$, $M=R/I$, where $I=(x_1)$, and $N=R$.

However, there are some trivial cases when $\operatorname{Hom}_R(M,N)$ is finitely generated, e.g. $M$ finitely generated and projective and $N$ finitely generated.