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(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:
- Any quotient of a cotorsion group is cotorsion.
- A direct product of groups is cotorsion if and only if each factor is.
- Every divisible group or injective group is cotorsion.
- The Baer Fomin Theorem states that a torsion group is cotorsion if and only if it is a direct sum of a divisible group and a bounded group, that is, a group of bounded exponent.
- A torsion-free abelian group is cotorsion if and only if it is algebraically compact.
- Ulm subgroups of cotorsion groups are cotorsion and Ulm factors of cotorsion groups are algebraically compact.