Definability of completely decomposable torsion-free Abelian groups by homomorphism groups. (Q1889457)

In the 1970's, \textit{P. Hill} [J. Algebra 19, 379-383 (1971; Zbl 0228.20027)] and \textit{A. M. Sebel'din} [Gruppy Moduly 70-77 (1976)] established that there is no Abelian group \(C\) such that for any Abelian groups \(A\) and \(B\), \(\Hom(A,C)\cong\Hom(B,C)\) implies that \(A\cong B\). In fact, Sebel'din demonstrated two completely decomposable nonisomorphic Abelian groups \(A\) and \(B\) such that \(\Hom(A,C)\cong\Hom(B,C)\) for any Abelian group \(C\). In the current work, the authors investigate the property that \(\Hom(C,A)\cong\Hom(C,B)\) implies that \(A\cong B\). Let \(C\) be an Abelian group. A class \(\chi\) of Abelian groups is called a `\(_CH\)-class' if the relation \(\Hom(C,A)\cong\Hom(C,B)\) implies that \(A\cong B\) for any groups \(A,B\in\chi\). Let \(\mathcal T\) be the class of completely decomposable torsion-free Abelian groups. In the current paper, Beregovaya and Sebel'din three times offer necessary and sufficient conditions on a group \(C\in\mathcal T\) that will ensure that a subclass \(\chi\) of \(\mathcal T\) is a \(_CH\)-class. The first of these conditions is the following: Theorem 1. Let \(C\in\mathcal T\). The class \(\mathcal T\) is a \(_CH\)-class if and only if \(C\) satisfies the following conditions: (1) \(C\) has a direct summand isomorphic to the integers; (2) \(\Omega(C)\) contains idempotent types only; (3) \(C\) is of finite rank.