Let $R$ be a commutative ring. Let $M$, $N$, and $L$ be $R$-modules.
When do we have $\text{Hom}(M, N) \cong \text{Hom}(M, R) \otimes N$? I am looking for the most general condition you can think of.
For instance, does this hold when $M$ and $N$ are finitely generated and projective? Is flatness relevant? It definitely holds when $R$ is a field and $M$ and $N$ are finite dimensional.
More generally, when do we have $\text{Hom}(M , N \otimes L) \cong \text{Hom}(M, N) \otimes L$?