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 formula_1, this says that formula_2 for all torsion-free groups formula_3. It suffices to check the condition for formula_3 the group of rational numbers.
More generally, a module "M" over a ring "R" is said to be a cotorsion module if Ext1("F","M")=0 for all flat modules "F". This is equivalent to the definition for abelian groups (considered as modules over the ring Z of integers) because over Z flat modules are the same as torsion-free modules. 
Some properties of cotorsion groups: