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Some Combinatorial Principles for Solving Algebraic Problems

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Infinite Length Modules

Part of the book series: Trends in Mathematics ((TM))

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Abstract

This is an extension of my talk at the Bielefeld conference in September 1998, which offers various infinite combinatorial principles proved by model theorists in the last two decades. These theorems are either based on ordinary set theory, like Shelah’s Black Box or the Shelah Elevator or need additional set theoretic axioms like CH or GCH or more which hold in Gödel’s universe. They are designed for applications in different areas of mathematics, mainly for proving non-structure theorems closely related to tame or wild representation type. In any case we will give examples of recent work in algebra in order to illustrate how these methods can be useful to algebraists in solving problems related with infinite structures.

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References

  1. T. Becker, L. Fuchs, S. Shelah, Whitehead modules over domains, Forum Mathematicum 1 (1989), 53–68.

    MATH  MathSciNet  Google Scholar 

  2. S. Brenner, C. M. Ringel, Pathological modules over tame rings, J. London Math. Soc. (2) 14 (1976), 207–215.

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Casacuberta, J. L. Rodriguez, J.-Y. Tai, Localization of abelian Eilenberg-Mac Lane spaces of finite type, preprint (1997).

    Google Scholar 

  4. A.L.S. Corner, Every countable reduced torsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 13 (1963), 687–710.

    Article  MATH  MathSciNet  Google Scholar 

  5. A.L.S. Corner, Additive categories and a theorem of W. G. Leavitt, Bull. Amer. Math. Soc. 75 (1969), 78–82.

    Article  MATH  MathSciNet  Google Scholar 

  6. A.L.S. Corner, R. Göbel, Prescribing endomorphism algebras - A unified treatment, Proceed. London Math. Soc. (3) 50 (1985), 471–483.

    Google Scholar 

  7. K. Devlin, S. Shelah, A weak version of ◊. which follows from \({2^{{\aleph _0}}} < {2^\aleph }^{_1}, \) Israel J. Math. 29 (1978), 239–247.

    Article  MATH  MathSciNet  Google Scholar 

  8. S. E. Dickson, A torsion theory for abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223–235.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Dugas, R. Göbel Every cotorsion-free ring is an endomorphism ring, Proceed. London Math. Soc. 45 (1982), 319–336.

    Article  MATH  Google Scholar 

  10. M. Dugas, R. Göbel Every cotorsion-free algebra is an endomorphism algebra, Math. Zeitschr. 181 (1982), 451–470.

    Article  MATH  Google Scholar 

  11. M. Dugas, R. Göbel, On radicals and products, Pacific J. Math. 18 (1985), 70–104.

    Google Scholar 

  12. M. Dugas, R. Göbel, Solution of Philip Hall’s problem on the existence of complete locally finite p-groups and results on E.C. groups, J. of Algebra 159 (1993), 115–138.

    Article  MATH  Google Scholar 

  13. M. Dugas, R. Göbel, Automorphisms of torsion-free nilpotent groups of class two, Trans. Amer. Math. Soc. 332 (1992), 633–646.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Dugas, R. Göbel, Torsion-free nilpotent groups and E-modules, Arch. Math. 54 (1990), 340–351.

    Article  MATH  Google Scholar 

  15. M. Dugas, R. Göbel, All infinite groups are Galois groups over any field, Trans. Amer. Math. Soc. 304 (1987), 355–384.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Dugas, R. Göbel Automorphism groups of fields, manuscripta mathematica 85 (1994), 227–242.

    Article  MATH  Google Scholar 

  17. M. Dugas and R. Göbel, Automorphism groups of fields II, Commun. in Algebra, 25 (1997), 3777–3785.

    Article  MATH  Google Scholar 

  18. M. Dugas, R. Göbel, Endomorphism rings of B2-groups of infinite rank, Israel Journal Math. 101 (1997), 141–156.

    Article  MATH  Google Scholar 

  19. M. Dugas, R. Göbel, Classification of modules with two distinguished pure submodules and bounded quotients, Results in Math. 30 (1996), 264–275.

    MATH  Google Scholar 

  20. M. Dugas, R. Göbel, W. May, Free modules with two distinguished submodules, Commun. Algebra 25 (1997), 3473–3481.

    Article  MATH  Google Scholar 

  21. M. Dugas, A. Mader, and C. Vinsonhaler, Large E-rings exist, J. Algebra 108 (1987), 88–101.

    Article  MATH  MathSciNet  Google Scholar 

  22. P. Eklof, Modules with strange decomposition properties, this volume, Birkhäuser Verlag Basel 1999

    Google Scholar 

  23. P. Eklof, A. Mekler Almost free modules, Set-theoretic methods, North-Holland, Amsterdam 1990

    MATH  Google Scholar 

  24. S. Files, R. Göbel, Gauß’s theorem for two submodules, Mathematische Zeitschrift 228 (1998), 511–536.

    Article  MATH  MathSciNet  Google Scholar 

  25. B. Franzen, R. Göbel, Prescribing endomorphism algebras. The cotorsion-free case. Rend. Sem. Mat. Padova 80 (1989), 215–241.

    Google Scholar 

  26. L. Fuchs, Infinite abelian groups - Volume 1,2 Academic Press, New York (1970, 1972)

    Google Scholar 

  27. W. Gaschütz, Nicht abelsche p-Gruppen besitzen äussere p-Automorphismen, J. Algebra 4 (1966), 1–2.

    Article  MATH  MathSciNet  Google Scholar 

  28. R. Göbel, Cotorsion-free abelian groups with only small cotorsion images, Austral. Math. Soc. (Ser. A) 50 (1991), 243–247.

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Göbel, New aspects for two classical theorems on torsion splitting, Comm. Algebra 15 (1987), 2473–2495.

    Article  MATH  MathSciNet  Google Scholar 

  30. R. Göbel, B. Goldsmith, On almost-free modules over complete discrete valuation rings, Rendiconti Padova 86 (1991), 75–87.

    MATH  Google Scholar 

  31. R. Göbel, W. May, Independence in completions and endomorphism algebras, Forum Mathematicum 1 (1989), 215–226.

    Article  MATH  MathSciNet  Google Scholar 

  32. R. Göbel, W. May, Four submodules suffice for realizing algebras over commutative rings, J. Pure Appl. Algebra 65 (1990), 29–43.

    Article  MATH  MathSciNet  Google Scholar 

  33. R. Göbel, W. May, Endomorphism algebras of peak I-spaces over posets of infinite prinjective type, Trans. Amer. Math. Soc. 349 (1997), 3535–3567.

    Article  MATH  MathSciNet  Google Scholar 

  34. R. Göbel, A. Paras, Outer automorphism groups of countable metabelian groups, pp. 309–317, in Proceedings of the Dublin Conference on Abelian Groups, Birkhauser Verlag, Basel (1999)

    Google Scholar 

  35. R. Göbel, A. Paras, Outer automorphism groups of metabelian groups, to appear in Journal of Pure and Applied Algebra 2000

    Google Scholar 

  36. R. Göbel, A. Paras, Automorphisms of metabelian groups with trivial center, Illinois J. Math. 42 (1998), 333–346.

    MATH  MathSciNet  Google Scholar 

  37. R. Göbel, J. L. Rodríguez, S. Shelah, Infinite localizations of finite simple groups, submitted to Journal reine and angew. Mathematik 1999

    Google Scholar 

  38. R. Göbel, S. Shelah, On the existence of rigid N1-free abelian groups of cardinality N1, pp. 227–237 in Proceedings of the Padova Conference on Abelian Groups and Modules, Kluwer, London 1995

    Google Scholar 

  39. R. Göbel, S. Shelah, G.C.H. implies the existence of many rigid almost free abelian groups, in Abelian Groups and Modules, pp. 253–271 in Proceedings of the international conference at Colorado Springs 1995, Lecture Notes in Pure and Appl. Math. 182 Marcel Dekker, New York 1996

    Google Scholar 

  40. R. Löbel, S. Shelah, Almost free indecomposable modules, the local case, Canadian Journal 50 (1998), 719–738.

    Google Scholar 

  41. R. Göbel, S. Shelah, Cotorsion theories and splitters, to appear Trans. Amer. Math. Soc. (June, 2000)

    Google Scholar 

  42. R. Göbel, S. Shelah, Almost free splitters, Colloquium Mathematicum 81 (1999), 193–221

    MATH  MathSciNet  Google Scholar 

  43. R. Göbel, S. Shelah, Almost free splitters under negation of CH, new results and a correction, to be submitted to Colloquium Math. 2000

    Google Scholar 

  44. R. Göbel, D. Simson, Embeddings of Kronecker modules into the category of prinjective modules and the endomorphism ring problem, Colloquium Math. 75 (1998), 213–244.

    MATH  Google Scholar 

  45. R. Göbel, D. Simson, Rigid families and endomorphism algebras of Kronecker modules, Israel Journal of Math 110 (1999), 293–315.

    Article  MATH  Google Scholar 

  46. R. Göbel, J. Trlifaj, Cotilting and a hierarchy of almost cotorsion groups, Journal of Algebra 224 (2000), 110–122.

    Article  MATH  MathSciNet  Google Scholar 

  47. R. Göbel, B. Wald, P. Westphal, Groups of integer valued functions, Springer Lecture Notes in Math. 874 (1981), 161–178.

    Article  Google Scholar 

  48. J. Gregory, Higher Souslin trees and the generalized continuum hypothesis, Journ. Symb. Logic 41 (1976), 663–667.

    Article  MATH  MathSciNet  Google Scholar 

  49. P. A. Griffith, A solution of the splitting mixed problem of Baer, Trans. American Math. Soc. 139 (1969), 261–269.

    Article  MATH  MathSciNet  Google Scholar 

  50. P. Griffith, N n –free abelian groups, Quart. J. Math. (2) 23 (1972), 417–425.

    Article  MATH  MathSciNet  Google Scholar 

  51. B. Hart, C. Laflamme, S. Shelah, Models with second order properties V. A general principle, Annals Pure Appl. Logic, Annals of Pure and Appl. Logic 64 (1993) 169–194.

    MATH  MathSciNet  Google Scholar 

  52. G. Higman, Almost free groups, Proc. London Math. Soc. 1 (1951), 184–190.

    MathSciNet  Google Scholar 

  53. G. Higman, Some countably free groups, pp. 129–150 in Proceedings “Group Theory”, Singapore 1991.

    Google Scholar 

  54. J. Hausen, Automorphismen gesättigte Klassen abzahlbarer abelscher Gruppen, Studies on Abelian Groups, Springer, Berlin (1968), 147–181.

    Google Scholar 

  55. P. Hill, Modular group algebras and simply presented groups, to appear in ‘Proceddings of the Dublin Conference on Abelian Groups’, (eds. P. Eklof, R. Göbel), Birkhäuser Verlag, Basel 1999

    Google Scholar 

  56. W. Hodges, Building models by games, Stud. Texts 2, Cambr. Univ. Press 1985

    Google Scholar 

  57. T. Jech, Set Theory, Academic Press, New York (1978)

    Google Scholar 

  58. R. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229–308.

    Article  MATH  MathSciNet  Google Scholar 

  59. A. Kanamori, The higher infinite, Perspectives in Math. Logic, Springer, Berlin, Heidelberg, New York 1994

    Google Scholar 

  60. G. Karpilovsky, Commutative group algebras, Lecture Notes Pure and Appl. Math., 78, Marcel Dekker, New York 1983

    Google Scholar 

  61. O. Kerner, Elementary stones, Comm. Algebra, 22 (1994), 1797–1806.

    Article  MATH  MathSciNet  Google Scholar 

  62. J. Lennox, S. Stonehewer, Subnormal subgroups of groups, Oxford Mathem. Monographs 1987

    MATH  Google Scholar 

  63. R. C. Lyndon, P. E. Schupp, Combinatorial Group Theory, Springer Ergebnisberichte, Vol. 89 Berlin, Heidelberg, New York 1977

    Google Scholar 

  64. M. Magidor, S. Shelah, When does almost free imply free? (for groups, transversals, etc.), Journ. Amer. Math. Soc. 7 (4) (1994), 769–830.

    Article  MATH  MathSciNet  Google Scholar 

  65. F. Menegazzo and S. Stonehewer, On the automorphism groups of a nilpotent p-group, J. London Math. Soc. (2) 3 (1985), 272–276.

    Article  MathSciNet  Google Scholar 

  66. N.N., this should be carried out, unpublished

    Google Scholar 

  67. M. Prest, Model theory and modules, London Math. Soc. L.N. 130, Cambridge University Press 1988

    Book  MATH  Google Scholar 

  68. C. M. Ringel, Infinite-dimensional representations of finite dimensional hereditary algebras, Symposia Math. 23 (1979), 321–412.

    MathSciNet  Google Scholar 

  69. C. M. Ringel, Bricks in hereditary length categories, Resultate der Mathematik 6 (1983), 64–70.

    MATH  MathSciNet  Google Scholar 

  70. 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

    Google Scholar 

  71. P. Rothmeier, 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.

    Google Scholar 

  72. A. N. Rudakov, Helices and vector bundles, London Math. Soc. LNM 148

    Google Scholar 

  73. L. Salce Cotorsion theories for abelian groups, Symposia Math. 23 (1979), 11–32.

    MathSciNet  Google Scholar 

  74. P. Schultz, Self-splitting groups, Preprint series of the University of Western Australia at Perth (1980)

    Google Scholar 

  75. S. Shelah Infinite abelian groups, Whitehead problem and some constructions, Israel Journal Math. 18 (1974), 243–256.

    Article  MATH  Google Scholar 

  76. S. Shelah On uncountable abelian groups, Israel Journal Math. 32 (1979), 311–330.

    Article  MATH  Google Scholar 

  77. S. Shelah, On a problem of Kurosh, Jonsson groups, and applications, S. I. Adian, W. W. Boone, G. Higman, eds. Word Problems II. North-Holland Publ. Co. (1980), 373–394.

    Google Scholar 

  78. 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

    Google Scholar 

  79. S. Shelah, A combinatorial principle and endomorphism rings. I: On p—groups, Israel J. Math. 49 (1984), 239–257.

    Article  MATH  MathSciNet  Google Scholar 

  80. S. Shelah, On successors of singular cardinals, Logic Colloquium ‘78, 97 (1978) of Stud. Logic Foundations Math., 357–380, North Holland, Amsterdam-New York.

    Google Scholar 

  81. S. Shelah, Non Structure Theory, Oxford University Press (2000) in preparation

    Google Scholar 

  82. S. Shelah, Z. Spasojevic, A forcing axiom on strongly inaccessible cardinals making all uniformizations for all ladder systems lying on a fixed stationary set and applications for abelian groups, manuscript No. 587.

    Google Scholar 

  83. M. Sollert Unabhängigkeitsaussagen in der Algebra - illustriert an der Theorie der abelschen Gruppen, Staatsexamensarbeit, Essen (1981)

    Google Scholar 

  84. L. Strüngmann, Almost-Free E(R)-algebras over countable domains, Ph. D. Thesis, Essen 1998

    Google Scholar 

  85. L. Unger, Schur modules over wild, finite dimensional path algebras with three non isomorphic simple modules, Journal Pure Appl. Algebra 64 (1990), 205–222.

    Article  MATH  MathSciNet  Google Scholar 

  86. T. Wakamatsu, On modules with trivial self extension, Journal of Algebra 114 (1988), 106–114.

    Article  MATH  MathSciNet  Google Scholar 

  87. R. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699–719.

    Article  MATH  MathSciNet  Google Scholar 

  88. U. H. Webb, An elementary proof of Gaschütz’s theorem, Arch. Math. 35 (1980), 23–26.

    Article  MATH  Google Scholar 

  89. A. E. Zalesskii, A nilpotent p-group has an outer automorphism, Dokl. Akad. Nauk SSSR 196 (1971), 751–754; Soviet Math. Dokl. 12 (1971), 227–230.

    Google Scholar 

  90. M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), 149–213.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Fachbereich 6, Mathematik und Informatik, Universität Essen, 45117, Essen, Germany

    Rüdiger Göbel

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  1. Rüdiger Göbel
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  1. Fakultät für Mathematik, Universität Bielefeld, D-33501, Bielefeld, Germany

    Henning Krause  & Claus Michael Ringel  & 

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Göbel, R. (2000). Some Combinatorial Principles for Solving Algebraic Problems. In: Krause, H., Ringel, C.M. (eds) Infinite Length Modules. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8426-6_5

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  • DOI: https://doi.org/10.1007/978-3-0348-8426-6_5

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9562-0

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