Homological Algebra and Set Theory

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of <!-- MATH ${\operatorname{Ext}}_\Lambda ^1$ --> for rings of global dimension 1. Using Martin's Axiom, the necessity of these conditions is shown not to be a theorem of ZFC. Applications are given to abelian group theory, including a partial solution (assuming <!-- MATH ${\text{V}} = {\text{L}}$ --> ) to a problem of Baer on the splitting of abelian groups. Some independence results in abelian group theory are also proved.