# Algebraically compact group

- 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 group is complete in the $mathbb\{Z\}$ 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**

* [*http://www.springerlink.com/index/W3W06361813J347X.pdf On endomorphism rings of Abelian groups*]

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