The Kaplansky test problems for $\aleph_1$-separable groups

by Eklof and Shelah. [EkSh:625]
Proc American Math Soc, 1998
We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for aleph_1-separable abelian groups of cardinality aleph_1 . In fact, there is an aleph_1-separable abelian group M such that M is isomorphic to M oplus M oplus M but not to M oplus M .

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