Английская Википедия:Algebraically compact group
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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>\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.
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