I know that $M_F$ = $M \otimes F$ over $R$ module; $R$ is PID and $F$ is a field of a fraction. Show that for all $t \in M_F$ there exist $a \in R$ such that $at \in M$.
Say $t=x \otimes y$ where $x \in M$ , $ y \in F$ such that $y = \frac{a}{b}$ and $a,b \in R$.
Thus, $bt = b(x \otimes \frac{a}{b}) $. So $ bt = x \otimes a$ by properties of tensor $bt \in M$.