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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 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== * (On endomorphism rings of Abelian groups ) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「algebraically compact group」の詳細全文を読む スポンサード リンク
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