Cotorsion group
In abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsion-free group splits. If the group is, this says that for all torsion-free groups. It suffices to check the condition for the group of rational numbers.
More generally, a module M over a ring R is said to be a cotorsion module if Ext1=0 for all flat modules F. This is equivalent to the definition for abelian groups because over Z flat modules are the same as torsion-free modules.
Some properties of cotorsion groups:
