An integral domain and its field of fractions. I'm trying to solve the following problem:

Let $R$ be a integral domain which is not a field and $K$ its fractions field. Show that a non-zero module $R$-homomorphism from $K$ to $R$ does not exist.

But I get stuck, someone can give some hint?
Thanks a lot!

Suppose $a \in R$ is a non-invertible element. Suppose $f: K \longrightarrow R$ is an $R$-linear map such that $f(1) \neq 0$. Then
$$
af(1)f \left( \frac{1}{af(1)} \right) = f \left( \frac{af(1)}{af(1)} \right) = f(1)
$$
hence $1= a f \left( \frac{1}{af(1)} \right)$ and this contradicts that $a$ is not invertible.
So it must be $f(1)=0$. And now for all $x, y \in R \setminus 0$
$$
y f \left( \frac{x}{y} \right) = f(x) = x f(1) =0
$$
so $f$ must be $0$.
