Nonreduced Abelian Groups with $\mathrm{UA}$-Rings of Endomorphisms

A ring $K$ is a unique addition ring (a $\mathrm{UA}$-ring) if its multiplicative semigroup $(K,\,\cdot\,)$ can be equipped with a unique binary operation $+$ transforming this semigroup to a ring $(K,\,\cdot\,,+)$. An Abelian group is called an $\operatorname{End}$-$\mathrm{UA}$-group if its endomorphism ring is a $\mathrm{UA}$-ring. In the paper, we find $\operatorname{End}$-$\mathrm{UA}$-groups in the class of nonreduced Abelian groups.