Geodesic
On the Question of Definability of Homogeneously Decomposable Torsion-Free Abelian Groups by Their Homomorphism Groups and Endomorphism Rings
Matematičeskie zametki, Tome 108 (2020) no. 1, pp. 130-136.

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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).
Keywords: homogeneously decomposable torsion-free Abelian group, definability of Abelian groups, group of homomorphisms, endomorphism ring. 
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T. A. Pushkova; A. M. Sebel'din. On the Question of Definability of Homogeneously Decomposable Torsion-Free Abelian Groups by Their Homomorphism Groups and Endomorphism Rings. Matematičeskie zametki, Tome 108 (2020) no. 1, pp. 130-136. http://geodesic.mathdoc.fr/item/MZM_2020_108_1_a9/

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