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A generalization of separable torsion-free Abelian groups. (English) Zbl 0571.20048

A torsion-free abelian group G is said to be separable if each finite subset of G is contained in a direct summand D which is a direct sum of finitely many rank one groups. In this paper, the authors generalize the idea of separability by replacing the rank one groups in the definition of separable groups by members of a class \({\mathfrak C}\). Such groups are called \({\mathfrak C}\)-separable. Members of the class \({\mathfrak C}\) are finite rank torsion-free abelian groups whose endomorphism rings are all principal ideal domains and a direct summand of a direct sum of countably many members of \({\mathfrak C}\) is again a direct sum of members of \({\mathfrak C}\). The authors prove that a direct summand of a \({\mathfrak C}\)-separable group is again \({\mathfrak C}\)-separable, thus extending the classical theorem of L. Fuchs on summands of separable groups. A theorem of E. F. Cornelius jun. [J. Algebra 67, 476-478 (1980; Zbl 0448.20051)] says that for the separability of a group G it is sufficient to assume that every element of G is contained in a summand which is a direct sum of rank one groups. Under mild conditions on \({\mathfrak C}\), the authors prove Cornelius’ Theorem for \({\mathfrak C}\)-separable abelian groups.
Reviewer: K.M.Rangaswamy

MSC:

20K20 Torsion-free groups, infinite rank
20K25 Direct sums, direct products, etc. for abelian groups
20K27 Subgroups of abelian groups

Citations:

Zbl 0448.20051
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References:

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