On subdirect sums of Abelian torsion-free groups of rank 1. (Q2519207)

The author studies a class of rank 2 torsion-free Abelian groups, called `special groups'. A rank 2 torsion-free group \(G\) is special if it is a subdirect sum of \(\mathbb{Q}\oplus\mathbb{Q}\) with the inducing group \(\mathbb{Q}/\mathbb{Z}\) and there exists a basic element in \(G\) (i.e. a pair \((a,b)\in\mathbb{Q}\oplus\mathbb{Q}\) such that \(\ker\varphi_1=\langle a\rangle\) and \(\ker\varphi_2=\langle b\rangle\); here \(\varphi_i\) are the structural epimorphisms which correspond to subdirect products). In the main result of the paper (Theorem 15) the author proves that there is a 1-1 correspondence between the set of all special groups with a fixed basic element and the multiplicative group of unity elements of the ring of universal numbers \(\prod_{p\in\mathbb{P}}\widehat\mathbb{Z}_p\).