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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On endomorphism rings of $ B_1$-groups that are not $ B_2$-groups


Author: Lutz Strüngmann
Journal: Proc. Amer. Math. Soc. 137 (2009), 3657-3668
MSC (2000): Primary 20K15, 20K20, 20K35, 20K40; Secondary 18E99, 20J05
Published electronically: June 15, 2009
MathSciNet review: 2529872
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Abstract: Finite rank Butler groups are pure subgroups of completely decomposable groups of finite rank and were defined by M.C.R. Butler. Extending this concept to infinite rank groups, Bican and Salce gave various possible descriptions: A $ B_2$-group $ G$ is a union of an ascending chain of pure subgroups $ G_{\alpha}$ such that for every $ \alpha$ we have $ G_{\alpha+1}=G_{\alpha}+H_{\alpha}$ for some finite rank Butler group $ H_{\alpha}$. A $ B_1$-group is a torsion-free group $ G$ satisfying $ \mathrm{Bext}_{\mathbb{Z}}^1(G,T)=0$ for all torsion groups $ T$. While the class of $ B_2$-groups is contained in the class of $ B_1$-groups, it is in general undecidable in ZFC if the two classes coincide. In this paper we study the endomorphism rings of $ B_1$-groups which are not $ B_2$-groups working in a model of ZFC that satisfies $ 2^{\aleph_0}=\aleph_4$.


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Additional Information

Lutz Strüngmann
Affiliation: Department of Mathematics, University of Duisburg-Essen, Campus Essen, 45117 Essen, Germany
Email: lutz.struengmann@uni-due.de

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09954-7
PII: S 0002-9939(09)09954-7
Received by editor(s): November 24, 2008
Received by editor(s) in revised form: March 3, 2009
Published electronically: June 15, 2009
Additional Notes: The author was supported by a grant from the German Research Foundation DFG
Communicated by: Julia Knight
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.