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 $$\mathbb{Z}$$ 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.