Abelian groups with UA-ring of endomorphisms and their homogeneous mappings

A ring $R$ is said to be a unique addition ring (UA-ring) if a multiplicative semigroup isomorphism $(R,{}^*)\cong(S,{}^*)$ is a ring isomorphism for any ring $S$. Moreover, a semigroup $(R,{}^*)$ is said to be a UA-ring if there exists a unique binary operation $+$ turning $(R,{}^*,+)$ into a ring. An $R$-module $A$ is called an $n$-endomorphal if any $R$-homogeneous mapping from $A^n$ to itself is linear. An $R$-module $A$ is called endomorphal if it is $n$-endomorphal for each positive integer $n$. In this paper, we consider the following classes of Abelian groups: torsion groups, torsion-free separable groups, and some indecomposable torsion-free groups of finite rank. We show that if an Abelian group is an endomorphal module over its endomorphism ring, then this ring is a UA-ring, and vice versa.