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An upper cardinal bound on absolute E-rings


Authors: Daniel Herden and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 137 (2009), 2843-2847
MSC (2000): Primary 20K30, 03E55, 03E75; Secondary 13C05, 03C25
DOI: https://doi.org/10.1090/S0002-9939-09-09842-6
Published electronically: February 24, 2009
MathSciNet review: 2506440
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that for every abelian group $ A$ of cardinality $ \ge\kappa(\omega)$ there exists a generic extension of the universe, where $ A$ is countable with $ 2^{\aleph_0}$ injective endomorphisms. As an immediate consequence of this result there are no absolute E-rings of cardinality $ \ge \kappa(\omega)$. This paper does not require any specific prior knowledge of forcing or model theory and can be considered accessible also for graduate students.


References [Enhancements On Off] (What's this?)

  • 1. M. Droste, R. Göbel, S. Pokutta, Absolute graphs with prescribed endomorphism monoid, Semigroup Forum 76(2) (2008), 256-267. MR 2377588 (2009a:20110)
  • 2. P. Eklof, A. Mekler, Almost Free Modules, Elsevier Science, North-Holland, Amsterdam (2002). MR 1914985 (2003e:20002)
  • 3. P. Eklof, S. Shelah, Absolutely rigid systems and absolutely indecomposable groups, Abelian Groups and Modules, Trends in Math., Birkhäuser, Basel (1999), 257-268. MR 1735574 (2001d:20050)
  • 4. R. Göbel, J. Trlifaj, Approximations and Endomorphism Algebras of Modules, Walter de Gruyter, Berlin (2006). MR 2251271 (2007m:16007)
  • 5. R. Göbel, S. Shelah, Absolutely indecomposable modules, Proc. Amer. Math. Soc. 135(6) (2007), 1641-1649. MR 2286071 (2007k:13047)
  • 6. R. Göbel, D. Herden, S. Shelah, Absolute E-rings (F805), in preparation.
  • 7. T. Jech, Set Theory, Springer-Verlag, Berlin (2003). MR 1940513 (2004g:03071)
  • 8. K. Kunen, Set Theory - An Introduction to Independence Proofs, North-Holland, Amsterdam (1980). MR 597342 (82f:03001)
  • 9. D. Marker, Model Theory: An Introduction, Springer-Verlag, New York (2002). MR 1924282 (2003e:03060)
  • 10. S. Shelah, Better quasi-orders for uncountable cardinals, Israel J. Math. 42(3) (1982), 177-226. MR 687127 (85b:03085)

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Additional Information

Daniel Herden
Affiliation: Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany
Email: Daniel.Herden@uni-due.de

Saharon Shelah
Affiliation: Einstein Institute of Mathematics, Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem 91904, Israel – and – Center for Mathematical Sciences Research, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
Email: Shelah@math.huji.ac.il

DOI: https://doi.org/10.1090/S0002-9939-09-09842-6
Received by editor(s): March 12, 2008
Received by editor(s) in revised form: November 23, 2008
Published electronically: February 24, 2009
Additional Notes: The first author was supported by a Wolfgang Gentner Minerva Fellowship.
The second author was supported by project No. I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research and Development.
Communicated by: Julia Knight
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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