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  In mathematics, in the realm of abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsionfree group splits. If the group is , this is equivalent to asserting that for all torsionfree groups . It suffices to check the condition for being 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 torsionfree modules.

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  In mathematics, in the realm of abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsionfree group splits. If the group is , this is equivalent to asserting that for all torsionfree groups . It suffices to check the condition for being 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 torsionfree 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 torsionfree 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.

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