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]].
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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 05:34, 9 February 2014

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