Cotorsion theories and splitters
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- by Rüdiger Göbel and Saharon Shelah
- Trans. Amer. Math. Soc. 352 (2000), 5357-5379
- DOI: https://doi.org/10.1090/S0002-9947-00-02475-2
- Published electronically: June 13, 2000
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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.References
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Bibliographic 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
- MR Author ID: 160185
- ORCID: 0000-0003-0462-3152
- Email: Shelah@math.huji.ae.il
- 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
- © Copyright 2000 American Mathematical Society
- 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
- DOI: https://doi.org/10.1090/S0002-9947-00-02475-2
- MathSciNet review: 1661246