Abstract
We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecomposable, that is, they remain indecomposable in any generic extension. However, any absolutely rigid family of groups has cardinality less than the partition cardinal κ(ω).
Partially supported by NSF Grants DMS-9501415 and DMS-9704477
Partially supported by NSF Grant DMS-9704477. Pub. No. 678
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Barwise, Back and forth through infinitary logic, in Studies in Model Theory (M. Morley, ed.), MAA (1973), 5–34.
J. Barwise and P. Eklof, Infinitary properties of Abelian torsion groups, Ann. Math. Logic 2 (1970), 25–68.
P. Eklof, Infinitary equivalence of abelian groups, Fund. Math. 81 (1974), 305–314.
L. Fuchs, Infinite Abelian Groups, Vols.I and II, Academic Press (1970, 1973).
L. Fuchs, Indecomposable abelian groups of measurable cardinalities, Symposia Math. XIII (1974), 233–244.
R. Göbel and W. May, Four submodules suffice for realizing algebras over commutative rings, J. Pure Appl. Algebra 65 (1990), 29–43.
T. Jech, Set Theory, Academic Press (1978).
C. Karp, Finite quantifier equivalence, in The Theory of Models (J. W. Addison et al, ed.), North-Holland (1965), 407–412.
R. Laver, On Fraïssé’s order type conjecture, Ann. Math. 93 (1971), 89–111.
R. Laver, Betterquasiordering s and a class of trees, in Studies in Foundations and Combinatorics (G.-C. Rota, ed.), Academic Press (1978), 31–48.
M. E. Nadel, Scott heights of abelian groups, J. Symb. Logic 59 (1994), 1351–1359.
C. St. J. A. Nash-Williams, On well-quasiordering infinite trees, Proc. Camb. Phil. Soc. 61 (1965), 697–720.
S. Shelah, Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math 18 (1974), 243–325.
S. Shelah, Better quasi-orders for uncountable cardinals, Israel J. Math. 42 (1982), 177–226.
J. Silver, A large cardinal in the constructible universe, Fund. Math. 69 (1970), 93–100.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this chapter
Cite this chapter
Eklof, P.C., Shelah, S. (1999). Absolutely rigid systems and absolutely indecomposable groups. In: Eklof, P.C., Göbel, R. (eds) Abelian Groups and Modules. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7591-2_21
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7591-2_21
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7593-6
Online ISBN: 978-3-0348-7591-2
eBook Packages: Springer Book Archive