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 ${\displaystyle M}$, this says that ${\displaystyle Ext(F,M)=0}$ for all torsion-free groups ${\displaystyle F}$. It suffices to check the condition for ${\displaystyle F}$ 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.