Completely decomposable homogeneous quotient divisible abelian groups

L.S. Pontryagin [1], A.G. Kurosh [2], A.I. Mal'cev [3], D. Derry [4], R. Baer [5], R. Beaumont and R. Pierce [6,7] began research on abelian torsion free groups of finite rank. In particular, R. Beaumont and R. Pierce [6] introduced the notion of the quotient divisible torsion free group. The notion of quotient divisible group was extended to the case of mixed groups in the work [8]. It was also proved in [8] that the category of quotient divisible mixed groups with quasi-homomorphisms was dual to the category of torsion-free finite-rank groups with quasi-homomorphisms. A modern version of the duality [8] was obtained in [9, 10]. The categories of groups with quasi-homomorphisms were replaced by the categories of groups with marked bases and with usual homomorphisms such that their matrices with respect to the marked bases consisted of integers. The duality [8] was also extended by S. Breaz and P. Schultz [11] on the class of self-small groups. The mixed quotient divisible groups as well as the self-small groups are in the focus of attention now [12-35].
In the present paper we prove two theorems about homogeneous completely decomposable quotient divisible mixed groups. In the first theorem we show that for every basis of such group there exists a decomposition of this group into a direct sum of rank-1 groups such that the elements of the basis are the bases of the corresponding rank-1 groups. Moreover, for every two bases such decompositions are isomorphic. In the second theorem we show that every exact sequence of quotient divisible groups $0\rightarrow B\rightarrow A\rightarrow C\rightarrow 0$ is splitting, if the group $A$ is homogeneous completely decomposable. This theorem is a dual version of the following classic result by R. Baer. Every pure subgroup of a homogeneous completely decomposable torsion free group of finite rank is a direct summand of this group.