# Cotorsion group

An Abelian group is a cotorsion group if for all torsion-free Abelian groups , i.e. every extension of by a torsion-free group splits (cf. also Extension of a group). For to be a cotorsion group it suffices to assume that . The importance of cotorsion groups lies in the facts that is a cotorsion group for all Abelian groups and , and that they have several nice features.

Cotorsion groups can also be characterized by their injective property with respect to those exact sequences of Abelian groups which split when is restricted to its torsion part (cf. also Exact sequence).

Epimorphic images of cotorsion groups are cotorsion, and so are the extensions of cotorsion groups by cotorsion groups. A direct product of groups is cotorsion if and only if each summand is cotorsion.

Examples of cotorsion groups are: 1) divisible (i.e., injective) Abelian groups, like , (cf. also Divisible group); and 2) algebraically compact groups, like finite groups and the additive group of the -adic integers (for any prime ); cf. also Compact group. A torsion Abelian group is cotorsion if and only if it is a direct sum of a divisible group and a bounded group (the Baer–Fomin theorem), and a torsion-free Abelian group is cotorsion exactly if it is algebraically compact. Ulm subgroups of cotorsion groups are cotorsion, and the Ulm factors of cotorsion groups are algebraically compact.

For a reduced cotorsion group , there is a natural isomorphism . This fact is relevant in showing that every Abelian group can be imbedded as a subgroup in a cotorsion group such that the cokernel is torsion-free and divisible. If is reduced, then can be chosen as ; this is the smallest cotorsion group in which can be imbedded in this manner. It is called the cotorsion hull of , and is unique up to isomorphism over .

A cotorsion group is said to be adjusted if it is reduced and contains no non-trivial torsion-free summand. The cotorsion hull of a reduced torsion group is adjusted, and the correspondence between the class of reduced torsion groups and the class of adjusted cotorsion groups is a bijection; its inverse is the formation of the torsion part. As a consequence, the classification of reduced torsion groups and that of adjusted cotorsion groups are equivalent problems. The Harrison structure theorem [a2] states that every cotorsion group is a direct sum of three groups: , where is a divisible group, is a reduced torsion-free algebraically compact group and is an adjusted cotorsion group. Such a decomposition of is unique up to isomorphism.

Some authors use "cotorsion" as "cotorsion in the above sense + reduced" .

A general reference is [a1]. See [a3] for a generalization to cotorsion modules over commutative domains.

See also Cotorsion-free group.

#### References

[a1] | L. Fuchs, "Infinite abelian groups" , 1 , Acad. Press (1970) |

[a2] | D.K. Harrison, "Infinite abelian groups and homological methods" Ann. of Math. , 69 (1959) pp. 366–391 |

[a3] | E. Matlis, "Cotorsion modules" , Memoirs , 49 , Amer. Math. Soc. (1964) |

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Cotorsion group.

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