Let $A$ be an abelian group, written additively, and endow $A$ with its canonical $\mathbb{Z}$-module structure, where for $n \in \mathbb{Z}$ and $a \in A$, the scalar product $na$ is defined by repeated addition, with $0a = 0_A$ and $(-n)a = -(na)$ for $n \in \mathbb{N}$. For a subset $B \subset A$, the subset $B$ is a $\mathbb{Z}$-submodule of $A$ if and only if $B$ is a subgroup of the additive group $A$.