## When is Ext(A,B) torsion-free?, and related problems.(English)Zbl 0653.20055

This paper considers the problem: Given torsion-free abelian groups A and B, when is Ext(A,B) torsion-free? The main result is the following: Theorem. Let A be countable, let B be finite rank with outer type of B not equal to the type of Q, the rational numbers, and let X be a subgroup of Q of type equal to the inner type of B. Then Ext(A,B) is torsion-free if and only if $$\cap \{Ker f|$$ f: $$A\to X\}$$ is divisible by all primes p such that pX$$\neq X.$$
This theorem extends an earlier result of Huber and Warfield for the case when B is rank one. The author also shows that under the hypotheses of the theorem, if Ext(A,B) is torsion-free, then there is a projective- injective relationship between A and B. Specifically, (1) B is injective with respect to any pure embedding $$0\to C\to A$$ where C is finite rank, and (2) A if projective with respect to any epimorphism $$B\to C\to 0$$ where C is torsion-free.
Reviewer: C.Vinsonhaler

### MSC:

 20K15 Torsion-free groups, finite rank 20K35 Extensions of abelian groups 20K40 Homological and categorical methods for abelian groups
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### References:

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