Prove that exits two $R-$modules $M,N$ different $0$ such that $Hom(M,N)=0$ Suppose that $R$ is a ring and $1_R$ is its multiplicative identity. Prove that exits two $R-$modules $M,N$ different $0$ such that $Hom(M,N)=0$??

That not true if $R$ is a field. Did you have anything else on your mind?
I think it is also not true for local artinian rings (say $\mathbb{Z}/p^n$ ).
It will be true for semisimple rings with two different classes of irreducible modules.
