Almost free splitters

Rüdiger Göbel; Saharon Shelah

Colloquium Mathematicae (1999)

  • Volume: 81, Issue: 2, page 193-221
  • ISSN: 0010-1354

Abstract

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Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that E x t R ( G , G ) = 0 . For simplicity we will call such modules splitters (see [10]). Also other names like stones are used (see a dictionary in Ringel’s paper [8]). Our investigation continues [5]. In [5] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [5] we concentrated on splitters which are larger than the continuum and such that countable submodules are not necessarily free. The “opposite” case of ℵ 1 -free splitters of cardinality less than or equal to ℵ 1 was singled out because of basically different techniques. This is the target of the present paper. If the splitter is countable, then it must be free over some subring of the rationals by Hausen [7]. In contrast to the results of [5] and in accordance with [7] we can show that all ℵ 1 -free splitters of cardinality ℵ 1 are free indeed.

How to cite

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Göbel, Rüdiger, and Shelah, Saharon. "Almost free splitters." Colloquium Mathematicae 81.2 (1999): 193-221. <http://eudml.org/doc/210735>.

@article{Göbel1999,
abstract = {Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that $Ext_R(G,G) = 0$. For simplicity we will call such modules splitters (see [10]). Also other names like stones are used (see a dictionary in Ringel’s paper [8]). Our investigation continues [5]. In [5] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [5] we concentrated on splitters which are larger than the continuum and such that countable submodules are not necessarily free. The “opposite” case of $ℵ_1$-free splitters of cardinality less than or equal to $ℵ_1$ was singled out because of basically different techniques. This is the target of the present paper. If the splitter is countable, then it must be free over some subring of the rationals by Hausen [7]. In contrast to the results of [5] and in accordance with [7] we can show that all $ℵ_1$-free splitters of cardinality $ℵ_1$ are free indeed.},
author = {Göbel, Rüdiger, Shelah, Saharon},
journal = {Colloquium Mathematicae},
keywords = {self-splitting modules; criteria for freeness of modules; splitters; torsion-free Abelian groups},
language = {eng},
number = {2},
pages = {193-221},
title = {Almost free splitters},
url = {http://eudml.org/doc/210735},
volume = {81},
year = {1999},
}

TY - JOUR
AU - Göbel, Rüdiger
AU - Shelah, Saharon
TI - Almost free splitters
JO - Colloquium Mathematicae
PY - 1999
VL - 81
IS - 2
SP - 193
EP - 221
AB - Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that $Ext_R(G,G) = 0$. For simplicity we will call such modules splitters (see [10]). Also other names like stones are used (see a dictionary in Ringel’s paper [8]). Our investigation continues [5]. In [5] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [5] we concentrated on splitters which are larger than the continuum and such that countable submodules are not necessarily free. The “opposite” case of $ℵ_1$-free splitters of cardinality less than or equal to $ℵ_1$ was singled out because of basically different techniques. This is the target of the present paper. If the splitter is countable, then it must be free over some subring of the rationals by Hausen [7]. In contrast to the results of [5] and in accordance with [7] we can show that all $ℵ_1$-free splitters of cardinality $ℵ_1$ are free indeed.
LA - eng
KW - self-splitting modules; criteria for freeness of modules; splitters; torsion-free Abelian groups
UR - http://eudml.org/doc/210735
ER -

References

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  1. [1] T. Becker, L. Fuchs and S. Shelah, Whitehead modules over domains, Forum Math. 1 (1989), 53-68. 
  2. [2] A. L. S. Corner and R. Göbel, Prescribing endomorphism algebras-A unified treatment, Proc. London Math. Soc. (3) 50 (1985), 471-483. Zbl0562.20030
  3. [3] P. Eklof and A. Mekler, Almost Free Modules. Set-Theoretic Methods, North-Holland, Amsterdam, 1990. Zbl0718.20027
  4. [4] L. Fuchs, Infinite Abelian Groups, Vols. 1, 2, Academic Press, New York, 1970, 1973. Zbl0209.05503
  5. [5] R. Göbel and S. Shelah, Cotorsion theories and splitters, Trans. Amer. Math. Soc. (1999), to appear. Zbl0962.20039
  6. [6] R. Göbel and J. Trlifaj, Cotilting and a hierarchy of almost cotorsion groups, J. Algebra (1999), to appear. Zbl0947.20036
  7. [7] J. Hausen, Automorphismen gesättigte Klassen abzählbaren abelscher Gruppen, in: Studies on Abelian Groups, Springer, Berlin, 1968, 147-181. 
  8. [8] C. M. Ringel, Bricks in hereditary length categories, Resultate Math. 6 (1983), 64-70. Zbl0526.16023
  9. [9] L. Salce, Cotorsion theories for abelian groups, Symposia Math. 23 (1979), 11-32. 
  10. [10] P. Schultz, Self-splitting groups, preprint, Univ. of Western Australia at Perth, 1980. 
  11. [11] S. Shelah, Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math. 18 (1974), 243-256. Zbl0318.02053
  12. [12] S. Shelah, On uncountable abelian groups, ibid. 32 (1979), 311-330. Zbl0412.20047
  13. [13] S. Shelah, A combinatorial theorem and endomorphism rings of abelian groups II, in: Abelian Groups and Modules, CISM Courses and Lectures 287, Springer, Wien, 1984, 37-86. 

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