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## Colloquium Mathematicum

1999 | 81 | 2 | 193-221
Tytuł artykułu

### Almost free splitters

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
193-221
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-08-27
poprawiono
1999-02-12
Twórcy
autor
• Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany
autor
• Department of Mathematics, Hebrew University, Jerusalem, Israel
Bibliografia
• [1] T. Becker, L. Fuchs and S. Shelah, Whitehead modules over domains, Forum Math. 1 (1989), 53-68.
• [2] A. L. S. Corner and R. Göbel, Prescribing endomorphism algebras-A unified treatment, Proc. London Math. Soc. (3) 50 (1985), 471-483.
• [3] P. Eklof and A. Mekler, Almost Free Modules. Set-Theoretic Methods, North-Holland, Amsterdam, 1990.
• [4] L. Fuchs, Infinite Abelian Groups, Vols. 1, 2, Academic Press, New York, 1970, 1973.
• [5] R. Göbel and S. Shelah, Cotorsion theories and splitters, Trans. Amer. Math. Soc. (1999), to appear.
• [6] R. Göbel and J. Trlifaj, Cotilting and a hierarchy of almost cotorsion groups, J. Algebra (1999), to appear.
• [7] J. Hausen, Automorphismen gesättigte Klassen abzählbaren abelscher Gruppen, in: Studies on Abelian Groups, Springer, Berlin, 1968, 147-181.
• [8] C. M. Ringel, Bricks in hereditary length categories, Resultate Math. 6 (1983), 64-70.
• [9] L. Salce, Cotorsion theories for abelian groups, Symposia Math. 23 (1979), 11-32.
• [10] P. Schultz, Self-splitting groups, preprint, Univ. of Western Australia at Perth, 1980.
• [11] S. Shelah, Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math. 18 (1974), 243-256.
• [12] S. Shelah, On uncountable abelian groups, ibid. 32 (1979), 311-330.
• [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.
Typ dokumentu
Bibliografia
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