## Absolutely indecomposable modules

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- by Rüdiger Göbel and Saharon Shelah
- Proc. Amer. Math. Soc.
**135**(2007), 1641-1649 - DOI: https://doi.org/10.1090/S0002-9939-07-08725-4
- Published electronically: January 8, 2007
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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 $\omega$-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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## Bibliographic Information

**Rüdiger Göbel**- Affiliation: Fachbereich 6, Mathematik, Universität Duisburg Essen, D 45117 Essen, Germany
- Email: r.goebel@uni-due.de
**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: shelah@math.huji.ac.il
- 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: https://doi.org/10.1090/S0002-9939-07-08725-4
- MathSciNet review: 2286071