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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Absolutely indecomposable modules
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by Rüdiger Göbel and Saharon Shelah PDF
Proc. Amer. Math. Soc. 135 (2007), 1641-1649 Request permission


A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more general result about $R$-modules over a large class of commutative rings $R$ with endomorphism ring $R$ which remains the same when passing to a generic extension of the universe. It turns out that ‘large’ in this context has a precise meaning, namely being smaller than the first $\omega$-Erdős cardinal defined below. We will first apply a result on large rigid valuated trees with a similar property established by Shelah in 1982, and will prove the existence of related ‘$R_\omega$-modules’ ($R$-modules with countably many distinguished submodules) and finally pass to $R$-modules. The passage through $R_\omega$-modules has the great advantage that the proofs become very transparent essentially using a few ‘linear algebra’ arguments also accessible for graduate students. The result closes a gap of Eklof and Shelah (1999) and Eklof and Mekler (2002), provides a good starting point for Fuchs and Göbel, and gives a new construction of indecomposable modules in general using a counting argument.
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Additional Information
  • Rüdiger Göbel
  • Affiliation: Fachbereich 6, Mathematik, Universität Duisburg Essen, D 45117 Essen, Germany
  • Email:
  • Saharon Shelah
  • Affiliation: Institute of Mathematics, Hebrew University, Jerusalem, Israel – and – Department of Mathematics, Rutgers University-New Brunswick, Piscataway, New Jersey 08854
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Email:
  • Received by editor(s): January 19, 2006
  • Received by editor(s) in revised form: February 19, 2006
  • Published electronically: January 8, 2007
  • Additional Notes: This work was supported by the project No. I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research & Development.
    This is GbSh880 in the second author’s list of publications.
  • Communicated by: Bernd Ulrich
  • © Copyright 2007 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 135 (2007), 1641-1649
  • MSC (2000): Primary 13C05, 13C10, 13C13, 20K15, 20K25, 20K30; Secondary 03E05, 03E35
  • DOI:
  • MathSciNet review: 2286071