Cotorsion group

In abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsion-free group splits. If the group is [math]\displaystyle{ M }[/math], this says that [math]\displaystyle{ Ext(F,M) = 0 }[/math] for all torsion-free groups [math]\displaystyle{ F }[/math]. It suffices to check the condition for [math]\displaystyle{ F }[/math] the group of rational numbers.

More generally, a module M over a ring R is said to be a cotorsion module if Ext¹(F,M)=0 for all flat modules F. This is equivalent to the definition for abelian groups (considered as modules over the ring Z of integers) because over Z flat modules are the same as torsion-free modules.

Some properties of cotorsion groups:

• Any quotient of a cotorsion group is cotorsion.
• A direct product of groups is cotorsion if and only if each factor is.
• Every divisible group or injective group is cotorsion.
• The Baer Fomin Theorem states that a torsion group is cotorsion if and only if it is a direct sum of a divisible group and a bounded group, that is, a group of bounded exponent.
• A torsion-free abelian group is cotorsion if and only if it is algebraically compact.
• Ulm subgroups of cotorsion groups are cotorsion and Ulm factors of cotorsion groups are algebraically compact.

External links

• Hazewinkel, Michiel, ed. (2001), "Cotorsion group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Cotorsion_group