Fuchs, L.; Viljoen, G. A generalization of separable torsion-free Abelian groups. (English) Zbl 0571.20048 Rend. Sem. Mat. Univ. Padova 73, 15-21 (1985). 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 Cited in 1 ReviewCited in 3 Documents MSC: 20K20 Torsion-free groups, infinite rank 20K25 Direct sums, direct products, etc. for abelian groups 20K27 Subgroups of abelian groups Keywords:torsion-free abelian group; separability; separable groups; finite rank torsion-free abelian groups; endomorphism rings; principal ideal domains; direct summand; \({\mathfrak C}\)-separable abelian groups Citations:Zbl 0448.20051 PDFBibTeX XMLCite \textit{L. Fuchs} and \textit{G. Viljoen}, Rend. Semin. Mat. Univ. Padova 73, 15--21 (1985; Zbl 0571.20048) Full Text: Numdam EuDML References: [1] D. Arnold - R. Hunter - F. Richman , Global Azumaya theorems in additive categories , J. Pure Appl. Algebra , 16 ( 1980 ), pp. 223 - 242 . MR 558485 | Zbl 0443.18014 · Zbl 0443.18014 · doi:10.1016/0022-4049(80)90026-2 [2] J.D. Botha - P.J. Gräbe , On torsion-free abelian groups whose endomorphism rings are principal ideal domains , Comm. in Alg. , Il ( 1983 ), pp. 1343 - 1354 . MR 697619 | Zbl 0515.20034 · Zbl 0515.20034 · doi:10.1080/00927878308822908 [3] B. Charles , Sous-groupes fonctoriels et topologies, Etudes sur les Groupes Abéliens , ed. B. Charles ( Dunod , Paris , 1968 ), pp. 75 - 92 . MR 240195 | Zbl 0179.32701 · Zbl 0179.32701 [4] E.F. Cornelius Jr., A sufficient condition for separability , J. Algebra , 67 ( 1980 ), pp. 476 - 478 . MR 602074 | Zbl 0448.20051 · Zbl 0448.20051 · doi:10.1016/0021-8693(80)90171-4 [5] L. Fuchs , Infinite abelian groups , Vol. I , Academic Press , New York and London , 1970 . MR 255673 | Zbl 0209.05503 · Zbl 0209.05503 [6] L. Fuchs , Infinite abelian groups , Vol. II , Academic Press , New York and London , 1973 . MR 349869 | Zbl 0257.20035 · Zbl 0257.20035 [7] C.E. Murley , The classification of certain classes of torsion-free abelian groups , Pacific J. Math. , 40 ( 1972 ), pp. 647 - 665 . Article | MR 322077 | Zbl 0261.20045 · Zbl 0261.20045 · doi:10.2140/pjm.1972.40.647 [8] K.M. Rangaswamy , On strongly balanced subgroups of separable torsionfree abelian groups, Abelian Group Theory , Lecture Notes in Math . 1006 ( 1983 ), pp. 268 - 274 . MR 722623 | Zbl 0574.20042 · Zbl 0574.20042 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.