In abelian group theory, an abelian group is said to be cotorsion if any extension of it by a torsion-free abelian group splits. If the group is , this says that the Ext-group is zero for all torsion-free groups . Since any such embeds into a direct sum of copies of , it suffices to check the condition on the group of rational numbers.
Every divisible group or injective group (in particular the group of rational numbers ) is cotorsion. For any two abelian groups and , the group of extensions is cotorsion.
Properties of cotorsion groups The class of cotorsion groups is closed under extensions, direct products, and quotients.
A countable cotorsion group is a direct sum of a divisible group and a bounded group, that is a group of bounded exponent.
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.
A torsion-free abelian group is cotorsion if and only if it is algebraically compact. Such groups are precisely the direct summands of direct products of -adic integers.
Ulm subgroups of cotorsion groups are cotorsion and Ulm factors of cotorsion groups are algebraically compact.
In ring theory A right module M over a ring R is said to be a cotorsion module if for all flat (right) modules F. When is the ring of integers , this reduces to the previous definition of cotorsion abelian groups.
The ring is said to be (right) cotorsion if the regular module is cotorsion.
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