Torsion-free Abelian groups with cyclic $p$-basic subgroups

We investigate the structure of indecomposable torsion-free Abelian groups all of whose $p$-basic subgroups are cyclic, and also the structure of the groups and rings of endomorphisms of such groups. We prove the existence of a torsion-free Abelian group of countable rank with cyclic $p$-basic subgroups which has no indecomposable nonzero direct summands.