## Absolutely indecomposable modules

HTML articles powered by AMS MathViewer

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

## 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.

## References

- Claudia Böttinger and Rüdiger Göbel,
*Endomorphism algebras of modules with distinguished partially ordered submodules over commutative rings*, J. Pure Appl. Algebra**76**(1991), no. 2, 121–141. MR**1145861**, DOI 10.1016/0022-4049(91)90055-7 - Gábor Braun and Rüdiger Göbel,
*Outer automorphisms of locally finite $p$-groups*, J. Algebra**264**(2003), no. 1, 55–67. MR**1980685**, DOI 10.1016/S0021-8693(03)00119-4 - Sheila Brenner,
*Endomorphism algebras of vector spaces with distinguished sets of subspaces*, J. Algebra**6**(1967), 100–114. MR**209319**, DOI 10.1016/0021-8693(67)90016-6 - Sheila Brenner and M. C. R. Butler,
*Endomorphism rings of vector spaces and torsion free abelian groups*, J. London Math. Soc.**40**(1965), 183–187. MR**174593**, DOI 10.1112/jlms/s1-40.1.183 - A. L. S. Corner,
*Endomorphism algebras of large modules with distinguished submodules*, J. Algebra**11**(1969), 155–185. MR**237557**, DOI 10.1016/0021-8693(69)90052-0 - A. L. S. Corner,
*Fully rigid systems of modules*, Rend. Sem. Mat. Univ. Padova**82**(1989), 55–66 (1990). MR**1049584** - A. L. S. Corner and Rüdiger Göbel,
*Prescribing endomorphism algebras, a unified treatment*, Proc. London Math. Soc. (3)**50**(1985), no. 3, 447–479. MR**779399**, DOI 10.1112/plms/s3-50.3.447 - A. L. S. Corner and Rüdiger Göbel,
*Small almost free modules with prescribed topological endomorphism rings*, Rend. Sem. Mat. Univ. Padova**109**(2003), 217–234. MR**1997988** - Manfred Dugas and Rüdiger Göbel,
*Automorphism groups of fields. II*, Comm. Algebra**25**(1997), no. 12, 3777–3785. MR**1481565**, DOI 10.1080/00927879708826085 - Manfred Dugas and Rüdiger Göbel,
*Automorphism groups of geometric lattices*, Algebra Universalis**45**(2001), no. 4, 425–433. MR**1816977**, DOI 10.1007/s000120050223 - Paul C. Eklof and Alan H. Mekler,
*Almost free modules*, Revised edition, North-Holland Mathematical Library, vol. 65, North-Holland Publishing Co., Amsterdam, 2002. Set-theoretic methods. MR**1914985** - Paul C. Eklof and Saharon Shelah,
*Absolutely rigid systems and absolutely indecomposable groups*, Abelian groups and modules (Dublin, 1998) Trends Math., Birkhäuser, Basel, 1999, pp. 257–268. MR**1735574** - Berthold Franzen and Rüdiger Göbel,
*The Brenner-Butler-Corner theorem and its applications to modules*, Abelian group theory (Oberwolfach, 1985) Gordon and Breach, New York, 1987, pp. 209–227. MR**1011314** - E. Fried and J. Kollár,
*Automorphism groups of fields*, Universal algebra (Esztergom, 1977) Colloq. Math. Soc. János Bolyai, vol. 29, North-Holland, Amsterdam-New York, 1982, pp. 293–303. MR**660867** - László Fuchs,
*Infinite abelian groups. Vol. II*, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR**0349869** - L. Fuchs and R. Göbel, Modules with absolute endomorphism rings, submitted.
- Rüdiger Göbel,
*Vector spaces with five distinguished subspaces*, Results Math.**11**(1987), no. 3-4, 211–228. MR**897298**, DOI 10.1007/BF03323270 - Rüdiger Göbel and Warren May,
*Four submodules suffice for realizing algebras over commutative rings*, J. Pure Appl. Algebra**65**(1990), no. 1, 29–43. MR**1065061**, DOI 10.1016/0022-4049(90)90098-3 - Rüdiger Göbel and Warren May,
*Independence in completions and endomorphism algebras*, Forum Math.**1**(1989), no. 3, 215–226. MR**1005423**, DOI 10.1515/form.1989.1.215 - Rüdiger Göbel and Saharon Shelah,
*Indecomposable almost free modules—the local case*, Canad. J. Math.**50**(1998), no. 4, 719–738. MR**1638607**, DOI 10.4153/CJM-1998-039-7 - R. Göbel and J. Trlifaj,
*Endomorphism Algebras and Approximations of Modules*, Walter de Gruyter Verlag, Berlin, Expositions in Mathematics, Vol. 41 (2006). - Hermann Heineken,
*Automorphism groups of torsionfree nilpotent groups of class two*, Symposia Mathematica, Vol. XVII (Convegno sui Gruppi Infiniti, INDAM, Roma, 1973) Academic Press, London, 1976, pp. 235–250. MR**0419627** - Thomas Jech,
*Set theory*, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**506523** - C. St. J. A. Nash-Williams,
*On well-quasi-ordering infinite trees*, Proc. Cambridge Philos. Soc.**61**(1965), 697–720. MR**175814**, DOI 10.1017/s0305004100039062 - Saharon Shelah,
*Infinite abelian groups, Whitehead problem and some constructions*, Israel J. Math.**18**(1974), 243–256. MR**357114**, DOI 10.1007/BF02757281 - Saharon Shelah,
*Better quasi-orders for uncountable cardinals*, Israel J. Math.**42**(1982), no. 3, 177–226. MR**687127**, DOI 10.1007/BF02802723 - Daniel Simson,
*Linear representations of partially ordered sets and vector space categories*, Algebra, Logic and Applications, vol. 4, Gordon and Breach Science Publishers, Montreux, 1992. MR**1241646**

## Additional 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