I am trying to work through the following problem:
Let $R$ be a principal ideal domain and let $K$ be the field of fractions of $R$. Let $M$ and $N$ be finitely generated $R$-modules. Let $Hom_R (M,N)$ be the set of $R$-module homomorphisms $\phi$ from $M$ to $N$. Show that
$$Hom_R (M, N) \otimes_R K \cong Hom_K (M \otimes_R K, N \otimes_R K)$$
My approach was to try to think of a trivial mapping between the two and then prove it is an isomorphism: $\psi_m (\phi \otimes k) = \phi(m) \otimes k$ but am not sure how to leverage the fact that both modules are finitely generated. Do I instead need to show we can map bases to bases?