Let $A$ be an abelian group, let $R$ be a commutative ring. $A$ is an $R$-module if it comes with an abelian group homomorphism $\alpha:R \to (A \to A)$ such that
$\alpha(1) = id_A$
for all $a:R$ and $b:R$, $\alpha(a) \circ \alpha(b) = \alpha(a \cdot b)$
Every abelian group is a $\mathbb{Z}$-module.