I have a set $\displaystyle S\subset{\mathbb{Q}}$ that is closed under addition and multiplication. Additionally, for any $\displaystyle r\in{\mathbb{Q}}$, one of the three is true

$\displaystyle r\in{S}, -r\in{S}, r=0$

I have to prove that all positive integers are in $\displaystyle S$.

So I assume there exists a positive integer $\displaystyle z\not\in{S}$, hence $\displaystyle -z\in{S}$.

Not sure which direction to go in to get the contradiction.

Help please?