give an example of Hom(M,N) and Hom(N,M) are not isomorphic as R modules

Question

give an example of a commutative unitary ring R and two finitely generated R modules M and N such that Hom(M,N) and Hom(N,M) are not isomorphic as R modules.

Answer 1

Try $R=\mathbb{Z}$, $M=\mathbb{Z}$, $N=\mathbb{Z}_2$. The first $Hom$ is non-trivial, while the second is trivial.