The Kaplansky test problems for aleph₁-separable groups

1997/09/15 by Paul C. Eklof, Saharon Shelah, Eklof, Paul C. +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rings and Algebras (math.RA) #Topological and Geometric Data Analysis #math.LO #math.RA

paper · pdf · doi:10.48550/arxiv.math/9709230

published as Proc. Amer. Math. Soc. 126 No. 7 (1998) 1901--1907

arxiv created 1997/09/15 · openalex publication_date 1997/09/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for aleph₁-separable abelian groups of cardinality aleph₁. In fact, there is an aleph₁-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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