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**Cobalanced exact sequences.**
*(English)*
Zbl 0693.20055

An exact sequence of abelian groups E: \(0\to A\to B\to C\to 0\) with C torsion free is balanced if every subgroup of the additive rationals Q is projective with respect to E. Dually, if A is torsion-free, E is called cobalanced if every subgroup of Q is injective with respect to E. A well- known result of Bican and Salce says that a torsion-free finite rank group C is a Butler group (pure subgroup of a finite rank completely decomposable group) if and only if \(Bext(C,T)=0\) for all torsion groups T. This last condition means that every balanced sequence E, with A torsion, is a split exact sequence.

In this note the authors study those torsion-free groups A such that \(Cobext(T,A)=0\) for all torsion T (every cobalanced sequence E, with C torsion is split exact). In general, such an A is characterized by the condition: A is isomorphic to a pure subgroup of a product of subgroups of Q. In the countable case, an equivalent condition is that A is locally completely decomposable.

In this note the authors study those torsion-free groups A such that \(Cobext(T,A)=0\) for all torsion T (every cobalanced sequence E, with C torsion is split exact). In general, such an A is characterized by the condition: A is isomorphic to a pure subgroup of a product of subgroups of Q. In the countable case, an equivalent condition is that A is locally completely decomposable.

Reviewer: C.Vinsonhaler

### MSC:

20K40 | Homological and categorical methods for abelian groups |

20K15 | Torsion-free groups, finite rank |

20K20 | Torsion-free groups, infinite rank |

20K27 | Subgroups of abelian groups |