Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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 [Enhancements On Off] (What's this?)

  • 1. T. Becker, L. Fuchs, S. Shelah, Whitehead modules over domains, Forum Mathematicum 1 (1989), 53-68. MR 90a:13017
  • 2. A.L.S. Corner, R. Göbel, Prescribing endomorphism algebras - A unified treatment, Proceed. London Math. Soc. (3) 50 (1985), 447-479. MR 86h:16031
  • 3. S. E. Dickson, A torsion theory for abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223-235. MR 35:162
  • 4. P. Eklof, A. Mekler, Almost free modules, Set-theoretic methods, North-Holland, Amsterdam 1990. MR 92e:20001
  • 5. B. Franzen, R. Göbel, Presribing endomorphism algeras. The cotorsion-free case. Rend. Sem. Mat. Padova 80 (1989), 215 - 241. MR 90h:16050
  • 6. L. Fuchs, Infinite abelian groups - Volume 1,2, Academic Press, New York (1970, 1972) MR 41:333; MR 50:2362
  • 7. P. A. Griffith, A solution of the splitting mixed problem of Baer, Trans. American Math. Soc. 139 (1969), 261-269. MR 39:317
  • 8. R. Göbel, Abelian groups with only small cotorsion images, J. Austral. Math. Soc. (Ser. A) 50 (1991), 243-247. MR 91m:20075
  • 9. R. Göbel, New aspects for two classical theorems on torsion splitting, Comm. Algebra 15 (1987), 2473-2495. MR 89h:13019
  • 10. R. Göbel, R. Prelle, Solution of two problems on cotorsion abelian groups, Archiv der Math. 31 (1978), 423-431. MR 80f:20063
  • 11. R. Göbel, S. Shelah, Almost free splitters, Colloq. Math. 81 (1999), 193-221.
  • 12. R. Göbel, J. Trlifaj, Cotilting and a hierarchy of almost cotorsion groups, to appear in Proc. Amer. Math. Soc.
  • 13. J. Hausen, Automorphismen gesättigte Klassen abzählbarer abelschen Gruppen, Studies on Abelian Groups, Springer, Berlin (1968), 147-181. MR 39:5691
  • 14. T. Jech, Set Theory, Academic Press, New York (1978). MR 80a:03062
  • 15. O. Kerner, Elementary stones, Comm. Algebra, 22 (1994) 1797 - 1806. MR 95d:16017
  • 16. M. Prest, Model theory and modules, London Math. Soc. L.N. 130 Cambridge University Press 1988. MR 89h:034061
  • 17. C. M. Ringel, The braid group action on the set of exceptional sequences of a hereditary artin algebra, pp. 339 - 352 in Abelian group theory and related topics, Contemporary Math. 171, American Math. Soc., Providence, R.I. 1994. MR 95m:16006
  • 18. C. M. Ringel, Bricks in hereditary length categories, Resultate der Mathematik 6 (1983) 64-70. MR 85a:16035
  • 19. P. Rothmaler, Purity in model theory, pp. 445 - 469 in Advances in Algebra and Model Theory, Series Algebra, Logic and Applications, Vol. 9, Gordon and Breach, Amsterdam 1997. CMP 99:12
  • 20. A. N. Rudakov, Helices and vector bundles, London Math. Soc. Lecture Note Ser. LNM 148. MR 91e:14002
  • 21. L. Salce, Cotorsion theories for abelian groups, Symposia Math. 23 (1979), 11-32. MR 81j:20078
  • 22. P. Schultz, Self-splitting groups, Preprint series of the University of Western Australia at Perth (1980).
  • 23. S. Shelah, A combinatorial theorem and endomorphism rings of abelian groups II, pp. 37 - 86, in Abelian groups and modules, CISM Courses and Lectures, 287, Springer, Wien 1984. MR 86i:20075
  • 24. L. Unger, Schur modules over wild, finite dimensional path algebras with three simple modules, Journal Pure Appl. Algebra 64 (1990) 205 - 222. MR 91i:16034
  • 25. T. Wakamatsu, On modules with trivial self-extensions, Journal of Algebra 114 (1988), 106-114. MR 89b:16020
  • 26. R. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969) 699 - 719. MR 39:4212
  • 27. M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984) 149 - 213. MR 86c:03034

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13D30, 18E40, 18G05, 20K20, 20K35, 20K40, 03C60, 18G25, 20K35, 20K40, 20K30, 13C10

Retrieve articles in all journals with MSC (2000): 13D30, 18E40, 18G05, 20K20, 20K35, 20K40, 03C60, 18G25, 20K35, 20K40, 20K30, 13C10

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

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

American Mathematical Society