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The Kaplansky test problems
for $\aleph _1$-separable groups


Authors: Paul C. Eklof and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 126 (1998), 1901-1907
MSC (1991): Primary 20K20; Secondary 03E35
DOI: https://doi.org/10.1090/S0002-9939-98-04749-2
MathSciNet review: 1485469
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Abstract: 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 $M\oplus M$. We also derive some relevant information about the endomorphism ring of $M$.


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Additional Information

Paul C. Eklof
Affiliation: Department of Mathematics, University of California, Irvine, California 92697
Email: peklof@math.uci.edu

Saharon Shelah
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Email: shelah@math.huji.ac.il

DOI: https://doi.org/10.1090/S0002-9939-98-04749-2
Keywords: Kaplansky test problems, $\aleph_1$-separable group, endomorphism ring
Received by editor(s): December 10, 1996
Additional Notes: Travel supported by NSF Grant DMS-9501415.
Research supported by German-Israeli Foundation for Scientific Research & Development Grant No. G-294.081.06/93. Pub. No. 625.
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1998 American Mathematical Society

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