Algebraically compact group

In mathematics, in the realm of abelian group theory, a group is said to be algebraically compact if it is a direct summand of every abelian group containing it as a pure subgroup. Equivalent characterizations of algebraic compactness:

* The reduced part of the group is Hausdorff and complete in the [math]\displaystyle{ \mathbb{Z} }[/math] adic topology.
* The group is pure injective, that is, injective with respect to exact sequences where the embedding is as a pure subgroup.

Relations with other properties:

* A torsion-free group is cotorsion if and only if it is algebraically compact.
* Every injective group is algebraically compact.
* Ulm factors of cotorsion groups are algebraically compact.

## External links

* On endomorphism rings of Abelian groups

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