On the Question of Definability of Homogeneously Decomposable Torsion-Free Abelian Groups by Their Homomorphism Groups and Endomorphism Rings

Let $C$ be an Abelian group. A class $X$ of Abelian groups is called a $_CH $-class (a $_CEH$-class) if, for any groups $A$ and $B$ in the class $X$, the isomorphism of the groups $\operatorname{Hom}(C,A)$ and $\operatorname{Hom}(C,B)$ (the isomorphism of the endomorphism rings $E(A)$ and $E(B)$ and of the groups $\operatorname{Hom}(C,A)$ and $\operatorname{Hom}(C,B)$) implies the isomorphism of the groups $A$ and $B$. In the paper, we study conditions that must be satisfied by a vector group $C$ for some class of homogeneously decomposable torsion-free Abelian groups to be a $_CH$ class (Theorem 1), and also, for some $C$ in the class of vector groups, for some class of homogeneously decomposable torsion-free Abelian groups to be a $_CEH$-class (Theorem 2).