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

Abstract

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).