Subgroups of the Baer–Specker group with few endomorphisms but large dual

Andreas Blass; Rüdiger Göbel

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 1, page 19-29
  • ISSN: 0016-2736

Abstract

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Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group ℤ ℵ 0 with the following properties. Every endomorphism of G differs from a scalar multiplication by an endomorphism of finite rank. Yet G has uncountably many homomorphisms to ℤ.

How to cite

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Blass, Andreas, and Göbel, Rüdiger. "Subgroups of the Baer–Specker group with few endomorphisms but large dual." Fundamenta Mathematicae 149.1 (1996): 19-29. <http://eudml.org/doc/212105>.

@article{Blass1996,
abstract = {Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group $ℤ^\{ℵ_0\}$ with the following properties. Every endomorphism of G differs from a scalar multiplication by an endomorphism of finite rank. Yet G has uncountably many homomorphisms to ℤ.},
author = {Blass, Andreas, Göbel, Rüdiger},
journal = {Fundamenta Mathematicae},
keywords = {dual groups; endomorphism rings; endomorphisms of finite rank; subgroups of finite rank; pure subgroups; direct summands},
language = {eng},
number = {1},
pages = {19-29},
title = {Subgroups of the Baer–Specker group with few endomorphisms but large dual},
url = {http://eudml.org/doc/212105},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Blass, Andreas
AU - Göbel, Rüdiger
TI - Subgroups of the Baer–Specker group with few endomorphisms but large dual
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 1
SP - 19
EP - 29
AB - Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group $ℤ^{ℵ_0}$ with the following properties. Every endomorphism of G differs from a scalar multiplication by an endomorphism of finite rank. Yet G has uncountably many homomorphisms to ℤ.
LA - eng
KW - dual groups; endomorphism rings; endomorphisms of finite rank; subgroups of finite rank; pure subgroups; direct summands
UR - http://eudml.org/doc/212105
ER -

References

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  1. [1] A. Blass, Cardinal characteristics and the product of countably many infinite cyclic groups, J. Algebra 169 (1994), 512-540. Zbl0816.20047
  2. [2] A. Blass, Near coherence of filters, II: Applications to operator ideals, the Stone-Čech remainder of a half-line, order ideals of sequences, and slenderness of groups, Trans. Amer. Math. Soc. 300 (1987), 557-581. Zbl0647.03043
  3. [3] A. Blass and C. Laflamme, Consistency results about filters and the number of inequivalent growth types, J. Symbolic Logic 54 (1989), 50-56. Zbl0673.03038
  4. [4] S. U. Chase, Function topologies on abelian groups, Illinois J. Math. 7 (1963), 593-608. Zbl0171.28703
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  7. [7] A. L. S. Corner and B. Goldsmith, On endomorphisms and automorphisms of some pure subgroups of the Baer-Specker group, in: Abelian Group Theory and Related Topics, R. Göbel, P. Hill and W. Liebert (eds.), Contemp. Math. 171, 1994, 69-78. Zbl0822.20056
  8. [8] M. Dugas und R. Göbel, Die Struktur kartesischer Produkte der ganzen Zahlen modulo kartesische Produkte ganzer Zahlen, Math. Z. 168 (1979), 15-21. Zbl0387.03018
  9. [9] M. Dugas und R. Göbel, Endomorphism rings of separable torsion-free abelian groups, Houston J. Math 11 (1985), 471-483. Zbl0597.20046
  10. [10] M. Dugas and J. Irwin, On basic subgroups of Π Z, Comm. Algebra 19 (1991), 2907-2921. Zbl0753.20015
  11. [11] M. Dugas and J. Irwin, On pure subgroups of cartesian products of integers, Resultate Math. 15 (1989), 35-52. Zbl0671.20052
  12. [12] M. Dugas, J. Irwin and S. Khabbaz, Countable rings as endomorphism rings, Quart. J. Math. Oxford 39 (1988), 201-211. Zbl0663.20058
  13. [13] K. Eda, A note on subgroups of ℤ ℕ , in: Abelian Group Theory, R. Göbel, L. Lady and A. Mader (eds.), Lecture Notes in Math. 1006, Springer, 1983, 371-374. 

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