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The Kaplansky test problems for -separable groups

Authors:	Paul C Eklof Saharon Shelah

Institution:	Department of Mathematics, University of California, Irvine, California 92697 ; Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

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 such that 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 .

Keywords:	Kaplansky test problems $\aleph_1$-separable group endomorphism ring

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