Proceedings of the American Mathematical Society

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



Absolutely indecomposable modules

Authors: Rüdiger Göbel and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 135 (2007), 1641-1649
MSC (2000): Primary 13C05, 13C10, 13C13, 20K15, 20K25, 20K30; Secondary 03E05, 03E35
Published electronically: January 8, 2007
MathSciNet review: 2286071
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Abstract: 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$-Erdos cardinal defined below. We will first apply a result on large rigid valuated trees with a similar property established by Shelah in 1982, and will prove the existence of related `$ 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

Saharon Shelah
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem, Israel – and – Department of Mathematics, Rutgers University-New Brunswick, Piscataway, New Jersey 08854

Keywords: Absolutely indecomposable modules, generic extension, distinguished submodules, labelled trees, Erd\H{o}s cardinal, rigid-like systems, automorphism groups.
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
Article copyright: © Copyright 2007 American Mathematical Society