Suppose $M$ is a $\mathbb{Q}$-module. Why is the given action of $\mathbb{Q}$ on $M$ (whatever it may be) the only way to make $M$ a $\mathbb{Q}$-module?
I know that an abelian group can only be made into a $\mathbb{Z}$-module in a unique way, since $1\cdot m=m$ for any $m\in M$. So for $p/q\in\mathbb{Q}$, $(p/q)\cdot m=\frac{1}{q}(pm)$. But $p\cdot m$ is necessarily the sum of $m$ a total of $p$ times. Also, $\frac{1}{q}(qm)=(\frac{1}{q}q)\cdot m=1\cdot m=m$. With this, does it somehow follow that there is only a unique way to make an abelian group $M$ a $\mathbb{Q}$-module when it is possible?