Periodic Abelian Groups with $UA$-Rings of Endomorphisms

A ring $R$ is said to be a unique addition ring (a $UA$-ring) if its multiplicative semigroup $(R,\cdot)$ can uniquely be endowed with a binary operation $+$ in such a way that $(R,\cdot,+)$ becomes a ring. An Abelian group is said to be an $\operatorname{End}$-$UA$-group if the endomorphism ring of the group is a $UA$-ring. In the paper we study conditions under which an Abelian group is an $\operatorname{End}$-$UA$-group.