Abelian Groups as $\mathrm{UA}$-Modules over the Ring $\mathbb Z$

Let $V$ be a module over a ring $R$. The module $V$ is called a unique addition module (a $\mathrm{UA}$-module) if there is no new addition on the set $V$ without changing the action of $R$ on $V$. In the paper, the $\mathrm{UA}$-modules over the ring $\mathbb Z$ are found.