Relative Prime Modulo Tensor is Zero

Theorem
Let $p \in \Z_{>0}$ and $q \in \Z_{>0}$ be positive coprime integers.

Let $\Z_p$ and $\Z_q$ be $\Z$-modules.

Then:
 * $\Z_p \otimes_\mathbb{Z} \Z_q = 0$

where $\otimes_\mathbb{Z}$ denotes tensor product.

Proof
By Bézout's Lemma there exists $a, b \in \Z$ such that $a p + b q = 1$.

Then for $s \otimes_\mathbb{Z} t \in \Z_p \otimes \Z_q$: