Homomorphism always exists between modules over an integral domain 
Let $R$ be an integral domain. Let $F$ be a free module over $R$, and let $M$ be an arbitrary nonzero $R$-module. Is it true that there always exists a nonzero module homomorphism from $M$ to $F$? 

I know that there always exists one from $F$ to $M$ by the universal property, but I don't know if it is true the other way around, and I can't think of any counter examples.

Not necessarily. Consider $R=\mathbb{Z}$, $F=\mathbb{Z}$, and $M = \mathbb{Z}/2\mathbb{Z}$.  If $\varphi: M \to F$ is a module homomorphism, then $2\varphi(1)=\varphi(2\cdot 1) = \varphi(0)=0$ implies that $\varphi(1)=0$, and hence there are no non-zero module homomorphisms from $M$ into $F$.
