Extensions in the class of countable torsion-free Abelian groups

It is a classical result that for a torsion-free Abelian group A the group is divisible for any Abelian group B . Hence it is of the form for some uniquely determined cardinals r 0 and r p . In this paper we clarify when and examine the possible values for r 0 and r p in case the groups A and B are countable (torsion-free). We also give some methods for constructing torsion-free groups A and B with prescribed cardinals r 0 and r p . This is to say that for suitable sequences ( r 0 , r p ∣ p ∈ℙ) of cardinals we construct torsion-free countable Abelian groups A and B realizing r 0 and r p as their invariants of .