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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 torsion-free group splits. If the group is , this is equivalent to asserting that for all torsion-free 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 torsion-free modules.

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  • Cotorsion group
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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 torsion-free group splits. If the group is , this is equivalent to asserting that for all torsion-free 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 torsion-free 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 torsion-free group splits. If the group is , this is equivalent to asserting that for all torsion-free 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 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.
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  • Cotorsion group
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