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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cotorsion theories and splitters


Authors: Rüdiger Göbel and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 352 (2000), 5357-5379
MSC (2000): Primary 13D30, 18E40, 18G05, 20K20, 20K35, 20K40; Secondary 03C60, 18G25, 20K35, 20K40, 20K30, 13C10
Published electronically: June 13, 2000
MathSciNet review: 1661246
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Abstract:

Let $R$ be a subring of the rationals. We want to investigate self splitting $R$-modules $G$ (that is $\operatorname{Ext}_R(G,G) = 0)$. Following Schultz, we call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples of splitters. Are there others? Answering an open problem posed by Schultz, we will show that there are more splitters, in fact we are able to prescribe their endomorphism $R$-algebras with a free $R$-module structure. As a by-product we are able to solve a problem of Salce, showing that all rational cotorsion theories have enough injectives and enough projectives. This is also basic for answering the flat-cover-conjecture.


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Additional Information

Rüdiger Göbel
Affiliation: Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117 Essen, Germany
Email: R.Goebel@Uni-Essen.De

Saharon Shelah
Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel, and Rutgers University, New Brunswick, New Jersey
Email: Shelah@math.huji.ae.il

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02475-2
PII: S 0002-9947(00)02475-2
Keywords: Cotorsion theories, completions, self-splitting modules, enough projectives, realizing rings as endomorphism rings of self-splitting modules. This paper is number GbSh 647 in Shelah's list of publications
Received by editor(s): February 23, 1998
Received by editor(s) in revised form: June 1, 1998, and November 18, 1998
Published electronically: June 13, 2000
Additional Notes: This work is supported by the project No. G-0294-081.06/93 of the German-Israeli Foundation for Scientific Research and Development
Article copyright: © Copyright 2000 American Mathematical Society