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n-FREE MODULES OVER COMPLETE DISCRETE VALUATION DOMAINS WITH ALMOST TRIVIAL DUAL*

Published online by Cambridge University Press:  25 February 2013

RÜDIGER GÖBEL
Affiliation:
Fakultät für Mathematik, Universität Duisburg-Essen Campus Essen, 45117 Essen, Germany e-mail: ruediger.goebel@uni-due.de
SAHARON SHELAH
Affiliation:
The Hebrew University, Givat Ram, Jerusalem 91904, Israel, and Rutgers University, New Brunswick, NJ 08901, USA e-mail: Shelah@math.huji.ac.il
LUTZ STRÜNGMANN
Affiliation:
Fakultät für Informatik, Hochschule Mannheim 68163 Mannheim, Germany e-mail: l.struengmann@hs-mannheim.de
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Abstract

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A module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the ℵn-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large ℵn-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013

Footnotes

*

Publication (GbShSm:981) in the second author's list of publications.

References

REFERENCES

1.Corner, A. L. S., Every countable reduced torsion-free ring is an endomorphism ring, Proc. Lond. Math. Soc. 13 (3) (1963), 687710.CrossRefGoogle Scholar
2.Corner, A. L. S. and Göbel, R., Prescribing endomorphism algebras – a unified treatment, Proc. Lond. Math. Soc. 50 (3) (1985), 447479.CrossRefGoogle Scholar
3.Dugas, M. and Göbel, R., Almost Σ-cyclic abelian p-groups in L, in Abelian groups and modules, proceedings of an international conference, Udine, CISM Courses and Lectures, vol. 287 (Springer, New York, 1984), 87105.Google Scholar
4.Eklof, P. C. and Mekler, A. H., Almost free modules, Revised ed. (North–Holland, New York, 2002).CrossRefGoogle Scholar
5.Fuchs, L., Infinite abelian groups, vol. 1 & 2 (Academic Press, New York, 1970, 1973).Google Scholar
6.Göbel, R., Herden, D. and Shelah, S., Prescribing endomorphism algebras of $\aleph_n$-free modules (to be submitted).Google Scholar
7.Göbel, R. and May, W., Independence in completions and endomorphism algebras, Forum Math. 1 (1989), 215226.Google Scholar
8.Göbel, R. and Paras, A., Splitting off free summands of torsion-free modules over complete DVRs, Glasgow Math. J. 44 (2002), 349351.CrossRefGoogle Scholar
9.Göbel, R. and Shelah, S., $\aleph_n$-free modules with trivial dual, Results Math. 54 (2009), 5364.Google Scholar
10.Göbel, R. and Trlifaj, J., Endomorphism algebras and approximations of modules, Expositions in Mathematics, vol. 41 (Walter de Gruyter Verlag, Berlin, Germany, 2006).CrossRefGoogle Scholar
11.Kaplansky, I., Infinite abelian groups (University of Michigan Press, Ann Arbor, MI, 1971).Google Scholar
12.Krylov, P. A. and Tuganbaev, A. A., Modules over discrete valuation domains, Expositions in Mathematics, vol 43 (Walter de Gruyter Verlag, Berlin, Germany, 2008).CrossRefGoogle Scholar
13.Shelah, S., $\aleph_n$-free abelian groups with no non-zero homomorphisms to ℤ, CUBO Math. J. 9 (2007), 5979.Google Scholar