Absolutely indecomposable modules

Rudiger Gobel*, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

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 ω-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 'Rω-modules' (R-modules with countably many distinguished submodules) and finally pass to R-modules. The passage through Rω-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.

Original languageEnglish
Pages (from-to)1641-1649
Number of pages9
JournalProceedings of the American Mathematical Society
Volume135
Issue number6
DOIs
StatePublished - Jun 2007

Keywords

  • Absolutely indecomposable modules
  • Automorphism groups
  • Distinguished submodules
  • Erdos cardinal
  • Generic extension
  • Labelled trees
  • Rigid-like systems

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