# Sh:519

- Göbel, R., & Shelah, S. (1995).
*On the existence of rigid \aleph_1-free abelian groups of cardinality \aleph_1*. In Abelian groups and modules (Padova, 1994), Vol. 343, Kluwer Acad. Publ., Dordrecht, pp. 227–237. arXiv: math/0104194 MR: 1378201 -
Abstract:

An abelian group is said to be \aleph_1–free if all its countable subgroups are free. Our main result is:If R is a ring with R^+ free and |R|<\lambda\leq 2^{\aleph_0}, then there exists an \aleph_1–free abelian group G of cardinality \lambda with {\rm End} G = R.

A corollary to this theorem is:

Indecomposable \aleph_1–free abelian groups of cardinality \aleph_1 do exist.

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Bib entry

@incollection{Sh:519, author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon}, title = {{On the existence of rigid $\aleph_1$-free abelian groups of cardinality $\aleph_1$}}, booktitle = {{Abelian groups and modules (Padova, 1994)}}, series = {Math. Appl.}, volume = {343}, year = {1995}, pages = {227--237}, mrclass = {20K20 (03C60 20K40)}, mrnumber = {1378201}, mrreviewer = {William J. Wickless}, publisher = {Kluwer Acad. Publ., Dordrecht}, note = {\href{https://arxiv.org/abs/math/0104194}{arXiv: math/0104194}}, arxiv_number = {math/0104194} }