The Kaplansky test problems for $\aleph _1$-separable groups
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- by Paul C. Eklof and Saharon Shelah PDF
- Proc. Amer. Math. Soc. 126 (1998), 1901-1907 Request permission
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 $M$.References
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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
- MR Author ID: 160185
- ORCID: 0000-0003-0462-3152
- Email: shelah@math.huji.ac.il
- 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
- © Copyright 1998 American Mathematical Society
- 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