Cotorsion group

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 ${\displaystyle M}$, this says that the Ext-group ${\displaystyle \operatorname {Ext} _{\mathbb {Z} }(F,M)}$ is zero for all torsion-free groups ${\displaystyle F}$. Since any such ${\displaystyle F}$ embeds into a direct sum of copies of ${\displaystyle \mathbb {Q} }$, it suffices to check the condition on the group of rational numbers.^[1]

Every divisible group or injective group (in particular the group of rational numbers ${\displaystyle \mathbb {Q} }$) is cotorsion. For any two abelian groups ${\displaystyle A}$ and ${\displaystyle B}$, the group of extensions ${\displaystyle \operatorname {Ext} _{\mathbb {Z} }(A,B)}$ is cotorsion.^[2]