On endomorphism rings of $B_1$-groups that are not $B_2$-groups
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- by Lutz Strüngmann
- Proc. Amer. Math. Soc. 137 (2009), 3657-3668
- DOI: https://doi.org/10.1090/S0002-9939-09-09954-7
- Published electronically: June 15, 2009
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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 $2^{\aleph _0}=\aleph _4$.References
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Bibliographic Information
- Lutz Strüngmann
- Affiliation: Department of Mathematics, University of Duisburg-Essen, Campus Essen, 45117 Essen, Germany
- Email: lutz.struengmann@uni-due.de
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 3657-3668
- MSC (2000): Primary 20K15, 20K20, 20K35, 20K40; Secondary 18E99, 20J05
- DOI: https://doi.org/10.1090/S0002-9939-09-09954-7
- MathSciNet review: 2529872