Show that any two elements of $M$ are linearly dependent over $\mathbb Z$.

Question

Let $M$ be the additive group of rationals .Show that any two elements of $M$ are linearly dependent over $\mathbb Z$.

I do not understand why is this true.If $c_1\dfrac{a}{b}+c_2\dfrac{c}{d}=0$ then $c_1=c_2=0$ .Is this result true or a fault in my arguement

Answer 1

Take any two distinct rationals $x_1={p_1\over q_1}$ and $x_2={p_2\over q_2}$ and consider the two following integers $\alpha_1=p_2q_1$ and $\alpha_2=-p_1q_2$. Those integers are not simultaneously zero because otherwise $x_1=x_2=0$. One obviously has $\alpha_1x_1+\alpha_2x_2=0$