Abelian Groups as $\mathrm{UA}$-Modules over Their Endomorphism Ring

Suppose that $V$ is a module over a ring $R$. The module $V$ is called a unique addition module ($\mathrm{UA}$-module) if it is not possible to change the addition on the set $V$ without changing the action of $R$ on $V$. In this paper, we find Abelian groups that are $\mathrm{UA}$-modules over their endomorphism ring.