Relative Prime Modulo Tensor is Zero

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

Let $\Z / p \Z$ and $\Z / q \Z$ be $\Z$-modules.

Then:
 * $\Z / p \Z \otimes_\Z \Z / q\Z = 0$

where $\otimes_\Z$ denotes tensor product over integers.

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

Then for $s \otimes_\Z t \in \Z / p \Z \otimes \Z / q \Z$:

Also see

 * Tensor Product of Quotient Rings, a generalization