On the existence of rigid $\aleph_1$-free abelian groups of cardinality $\aleph_1$

by Goebel and Shelah. [GbSh:519]
Abelian Groups and Modules, 1995
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 <= 2^{aleph_0}, then there exists an aleph_1 --free abelian group G of cardinality lambda with 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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