Relative Prime Modulo Tensor is Zero

Proposition
Let $\mathbb{Z}_p$ and $\mathbb{Z}_q$ be given modules such that $p$ and $q$ are relative prime.

Then we have that

$\mathbb{Z}_p\otimes \mathbb{Z}_q = 0$

Proof
By Bézout's Lemma does there exist $a,b\in\mathbb{Z}$ such that $ap+bq=1$.

Then for $s\otimes t\in \mathbb{Z}_p\otimes \mathbb{Z}_q$