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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 M {\displaystyle M} , this says that E x t = 0 {\displaystyle Ext=0} for all torsion-free groups F {\displaystyle F}. It suffices to check the condition for F {\displaystyle F} the group of rational numbers.
More generally, a module M over a ring R is said to be a cotorsion module if Ext=0 for all flat modules F. This is equivalent to the definition for abelian groups because over Z flat modules are the same as torsion-free modules.
Some properties of cotorsion groups:
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