# Sh:880

- Göbel, R., & Shelah, S. (2007).
*Absolutely indecomposable modules*. Proc. Amer. Math. Soc.,**135**(6), 1641–1649. arXiv: 0711.3011 DOI: 10.1090/S0002-9939-07-08725-4 MR: 2286071 -
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 the*precise meaning,*namely being smaller then the first \omega-Erdos cardinal defined below. We will first apply result on large rigid 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 accessible also for graduate students. The result gives a new construction of indecomposable modules in general using a counting argument. - published version (9p)

Bib entry

@article{Sh:880, author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon}, title = {{Absolutely indecomposable modules}}, journal = {Proc. Amer. Math. Soc.}, fjournal = {Proceedings of the American Mathematical Society}, volume = {135}, number = {6}, year = {2007}, pages = {1641--1649}, issn = {0002-9939}, doi = {10.1090/S0002-9939-07-08725-4}, mrclass = {13L05 (03E55 13C05 20K20 20K30)}, mrnumber = {2286071}, mrreviewer = {Luigi Salce}, doi = {10.1090/S0002-9939-07-08725-4}, note = {\href{https://arxiv.org/abs/0711.3011}{arXiv: 0711.3011}}, arxiv_number = {0711.3011} }