Extensions in the class of countable torsion-free Abelian groups

Abstract

It is a classical result that for a torsion-free Abelian group A the group \(\operatorname {Ext}_{\mathbb {Z}}(A,B)\) is divisible for any Abelian group B. Hence it is of the form Open image in new window for some uniquely determined cardinals r0 and rp. In this paper we clarify when \(\operatorname {Ext}_{\mathbb {Z}}(A,B)=0\) and examine the possible values for r0 and rp 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 r0 and rp. This is to say that for suitable sequences (r0,rp∣p∈ℙ) of cardinals we construct torsion-free countable Abelian groups A and B realizing r0 and rp as their invariants of \(\operatorname {Ext}_{\mathbb {Z}}(A,B)\).