Stewart's theorem: Difference between revisions
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In [[mathematics]], in the realm of [[Abelian group|abelian]] [[group theory]], a [[Group (mathematics)|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 <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 group|cotorsion]] if and only if it is algebraically compact. | |||
* Every [[injective group]] is algebraically compact. | |||
* [[Ulm factor]]s of cotorsion groups are algebraically compact. | |||
==External links== | |||
* [http://www.springerlink.com/index/W3W06361813J347X.pdf On endomorphism rings of Abelian groups] | |||
[[Category:Abelian group theory]] | |||
[[Category:Properties of groups]] | |||
Revision as of 11:36, 6 January 2014
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 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.