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, ċ) can uniquely be endowed with a binary operation + in such a way that (R, ċ, +) becomes a ring. An Abelian group is said to be an 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 End-UA-group.