Almost free splitters

• Autorzy: Rüdiger Göbel , Saharon Shelah
• 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

self-splitting modules   criteria for freeness of modules  

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1999
• Tom: 81
• Numer: 2
• Strony: 193-221

Daty

wydano

1999

otrzymano

1998-08-27

poprawiono

1999-02-12

Twórcy

autor

Rüdiger Göbel

• Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany

autor

Saharon Shelah

• 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.