Morita Duality — A Survey

  • Bruno J. Müller
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 287)


We leave aside the prolific and interesting investigations concerned with dualities between categories of topological modules, be they extensions or generalizations of Morita dualities, between more general categories, or between module categories with less perfect closure properties. Instead we confine ourselves totally to the original setting, as introduced by Morita himself, and defined in Section 1 . We feel that the core problems, regarding existence and selfduality in natural situations, present themselves most clearly here. Some progress has been made in recent years, mainly through the contributions of Schofield (division rings), Vamos (commutative and PI-rings), Haack and Dischinger — Müller (artinian serial rings), Jategaonkar, Musson and Donkin (noetherian rings), and MacDonald and Ann (generalized Morita dualities).


Prime Ideal Maximal Ideal Division Ring Polynomial Identity Noetherian Ring 
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  1. 1.
    Anh, Pham Ngoc, Duality over noetherian rings with a Morita duality, J. Algebra 75 (1982), 275–285.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Anh, Pham Ngoc, Duality of modules over topological rings, J. Algebra 75 (1982), 395–425.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Anh, Pham Ngoc, On a problem of B. J. Müller, Archiv Math. 39 (1982), 303–305.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Anderson, F. W. and K. R. Fuller, Rings and Categories of Modules, Springer Verlag 1974.CrossRefMATHGoogle Scholar
  5. 5.
    Auslander, M., M. I. Platzeck and I. Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979), 1–46.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Azumaya, G., A duality theory for infective modules,. Amer. J. Math. 81 (1959), 249–278.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Azumaya, G., Exact and serial rings, J. Algebra 85 (1983), 477–489.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Brown, K. A., T. H. Lenagan and J. T. Stafford, K.theory and stable structure of some noetherian group rings, Proc. London Math. Soc. 42 (1981), 193–230.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Brown, K. A., Localisation, bimodules and injective modules for enveloping algebras of solvable Lie algebras, Bull. Sci. Math. 107 (1983), 225–251.MATHGoogle Scholar
  10. 10.
    Brown, K. A., Ore sets in enveloping algelras, Comp. Math. (to appear).Google Scholar
  11. 11.
    Brown, K. A., Ore sets in noetherian rings (preprint).Google Scholar
  12. 12.
    Conn, P. M., Quadratic extensions of skew fields, Proc. London Math. Soc. 11 (1961), 531–556.Google Scholar
  13. 13.
    Cohn, P. M., On a class of binomial extensions, 111. J. Math. 10 (1966), 418–424.MATHGoogle Scholar
  14. 14.
    Cozzens, J. H., Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Eisenbud, D. and J. C. Robson, Hereditay noetherian prime rings, J. Algebra 16 (1970), 86–104.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Deshpande, V. K., Completions of noetherian hereditary prime rings, Pacific J. Math. 90 (1980), 285–297.MATHMathSciNetGoogle Scholar
  17. 17.
    Dieudonne, J., Remarks on quasi-Frobenius rings, I11. J. Math. 2 (1958), 346–354.MATHMathSciNetGoogle Scholar
  18. 18.
    Dischinger, F. and W. Müller, Die triviale Ringerweiterung R &BEB ist links PF aber nicht rechts PF (preprint).Google Scholar
  19. 19.
    Dischinger, F. and W. Müller, Einreihig zerlegbare artinsche Ringe sind selbstdual (preprint).Google Scholar
  20. 20.
    Donkin, S., Locally finite representations of polycyclic-by-finite groups, Proc. London Math. Soc. 44 (1982), 333–348.CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Fuller, K. R. and J. Haack, Rings with quivers that are trees, Pacific J. Math. 76 (1978), 371–379.MATHMathSciNetGoogle Scholar
  22. 22.
    Fuller, K. R. and J. K. Haack, Duality for semigroup rings, J. Pure Applied Algebra 22 (1981), 113–119.CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Goblot, R., Sur les anneaux lineairement compacts, G. R. Acad. Sci. Paris 270 (1970), A 1212–1215.MATHMathSciNetGoogle Scholar
  24. 24.
    Haack, J. K., Incidence rings with self-duality, Proc. Amer. Math. Soc. 78 (1980), 165–169.CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Haack, J. K., Self-duality and serial rings, J. Algebra 59 (1979), 345–363.CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Haack, J. K., Serial rings and sudirect products, J. Pure Applied Algebra (to appear).Google Scholar
  27. 27.
    Haack, J. K., V. P. Camillo and K. R. Fuller, Azumaya’s exact rings and a problem in linear algebra (preprint).Google Scholar
  28. 28.
    Hinohara, Y., Note on non-commutative semi-local rings, Nagoya J. Math. 17 (1960), 161–166.MATHMathSciNetGoogle Scholar
  29. 29.
    Jategaonkar, A. V., Injective modules and localization in noncommu-tative noetherian rings, Trans. Amer. Math. Soc. 188 (1974), 109–123.CrossRefMathSciNetGoogle Scholar
  30. 30.
    Jategaonkar, A. V., Integral group rings of polycyclic-by-finite groups, J. Pure Applied Algebra 4 (1974), 337–343.CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Jategaonkar, A. V., Certain injectives are artinian, Lecture Notes Math. 545 (1976), Springer Verlag, 128–139.CrossRefMathSciNetGoogle Scholar
  32. 32.
    Jategaonkar, A. V., Morita duality and noetherian rings, J. Algebra 69 (1981), 358–371.CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Jategaonkar, A. V., Localization in noetherian rings (preprint).Google Scholar
  34. 34.
    Kaplansky, I., On a problem of Kurosch and Jacobson, Bull. Amer. Math. Soc. 52 (1946), 496–500.CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Kaplansky, I., Dual modules over valuation rings, Proc. Amer. Math. Soc. 4 (1953), 213–219.CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Kitamura, Y., Quasi-Frobenius extensions with Morita duality, J. Algebra 73 (1981), 275–286.CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Lemonnier, B., AB5et la dualite de Morita, G. R. Acad. Sci. Paris 289 (1979), A 47–50.MATHMathSciNetGoogle Scholar
  38. 38.
    Macdonald, R. N. S., Dualities between finitely closed subcategories of modules, Ph.D. thesis, MacMaster University (1977).Google Scholar
  39. 39.
    Macdonald, R. N. S., Representable dualities between finitely closed subcategories of modules, Canad. J. Math. 31 (1979), 465–475.MATHMathSciNetGoogle Scholar
  40. 40.
    Marubayashi, H., Completions of hereditary noetherian prime rings, Osaka J. Math, 17 (I980), 391–406.MathSciNetGoogle Scholar
  41. 41.
    Matlis, E., Infective modules over noetherian rings, Pacific J. Math. 8 (1958), 511–528.MATHMathSciNetGoogle Scholar
  42. 42.
    McConnell, J. C., Localisation in enveloping rings, J. London Math. Soc. 4–3 (1968), 421–428.CrossRefMathSciNetGoogle Scholar
  43. 43.
    McConnell, J. C., The noetherian property in complete rings and modules, J. Algebra 12 (1969), 143–153.CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    McConnell, J. C., On completions of non-commutative noetherian rings, Comm. Algebra 6 (1978), 1485–1488.MATHMathSciNetGoogle Scholar
  45. 45.
    Morita, K., Duality for modules and its applications to the theory of rings with minimum condition, Sci. Report Tokyo Kyoiku Daigaku 6 (1958), 83–142.MATHGoogle Scholar
  46. 46.
    Müller, B. J., On Morita duality, Canad. J. Math 21 (1969), 1338–1347.MATHGoogle Scholar
  47. 47.
    Müller, B. J., Linear compactness and Morita duality, J. Algebra 16 (1970), 60–66.CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Müller, B. J., The structure of quasi-Frobenius rings, Canad. J. Math. 26 (1974), 1141–1151.MATHGoogle Scholar
  49. 49.
    Müller, B. J., Localization in non-commutative noetherian rings, Canad. J. Math. 28 (1976), 600–610.MATHGoogle Scholar
  50. 50.
    Müller, B. J., Localization in fully bounded noetherian rings, Pacific J. Math 67 (1976), 233–245.Google Scholar
  51. 51.
    Müller, B. J., Two-sided localization in noetherian PI-rings, J. Algebra 63 (l980), 359–373.CrossRefMathSciNetGoogle Scholar
  52. 52.
    Müller, B.J., Links between maximal ideals in bounded noetherian prime rings of Krull dimension one, Proc. Gonf. Methods in Ring Theory, Antwerp (1983) (to appear).Google Scholar
  53. 53.
    Müller, B. J., Affine noetherian PI-rings have enough clans, J. Algebra (to appear).Google Scholar
  54. 52.
    Müller, W., Unzerlegbare Moduln über artinschen Ringen, Math. Z. 137 (1974), 197–226.CrossRefMATHMathSciNetGoogle Scholar
  55. 55.
    Musson, I. M., I.fective modules for group rings in polycyclic groups I, Quart. J. Math. Oxford 31 (1980), 429–448.CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Osofsky, B. L., A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373–387.CrossRefMATHMathSciNetGoogle Scholar
  57. 57.
    Passman, D. S., Universal fields of fractions for polycyclic group algebras, Glasgow J. Math. 23 (1982), 103–113.CrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    Procesi, G., Rings with Polynomial Identities, Marcel Dekker (1973).MATHGoogle Scholar
  59. 59.
    Rosenberg, A. and D. Zelinsky, Finiteness of the infective hull, Math, Z. 70 (1959), 372–380.CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    Roux, B., Sur la dualite de Morita, Tohoku J. Math 23 (1971), 457–4–72.CrossRefMATHMathSciNetGoogle Scholar
  61. 61.
    Roux, B., Anneaux artiniens et extensions dfanneaux semi-simples, J. Algebra 25 (1973), 295–306.CrossRefMATHMathSciNetGoogle Scholar
  62. 62.
    Roux, B., Modules injectifs indecomposables sur les anneaux artiniens et dualite de Morita, Seminaire P. Dubreil 1972/73, Secretariat Math. Paris (1973), PP. 19.Google Scholar
  63. 63.
    Sandomierski, F. L., Linearly compact modules and local Morita duality, Proc. Gonf. Ring Theory Park City 1971, Academic Press (1972), 333–346.Google Scholar
  64. 64.
    Schofield, A. H., (I’ve heard rumors).Google Scholar
  65. 65.
    Stafford, J. T., The Goldie rank of a module (preprint).Google Scholar
  66. 66.
    Tachikawa, H., Duality theorem of character modules for rings with minimum condition, Math. Z. 68 (1958), 479–487.CrossRefMATHMathSciNetGoogle Scholar
  67. 67.
    Tachikawa, H., Quasi-Frobenius Rings and Generalizations, Lecture Notes in Math. 351, Springer Verlag (1973).MATHGoogle Scholar
  68. 68.
    Upham, M. H., Localization and completion of FBN hereditary rings, Comm. Algebra 7 (1979), 1269–1307.MATHMathSciNetGoogle Scholar
  69. 69.
    Upham, M. H., Two remarks on duality, Houston J. Math. 5 (1979), 437–443.MATHMathSciNetGoogle Scholar
  70. 70.
    Upham, M. H., The first and second endomorphism rings of an infective module over an FBN ring (prerpint).Google Scholar
  71. 71.
    Utumi, Y., Self-injective rings, J. Algebra 6 (1967), 56–64.CrossRefMATHMathSciNetGoogle Scholar
  72. 72.
    Vamos, P., Rings with duality, Proc. London Math. Soc. 35 (1977), 275–289.MATHMathSciNetGoogle Scholar
  73. 73.
    Vamos, P., Semi-local noetherian Pi-rings, Bull. London Math. Soc. 9 (1977), 251–256.CrossRefMATHMathSciNetGoogle Scholar
  74. 74.
    Wiseman, A. N., Integral extensions of linearly compact domains, Comm. Algebra 11 (1983), 1099–1121.MATHMathSciNetGoogle Scholar
  75. 75.
    Zelinsky, D., Linearly compact modules and rings, Amer. J. Math. 75 (1953), 79–90.CrossRefMATHMathSciNetGoogle Scholar
  76. 76.
    Zöschinger,H., Linear-kompakte Moduln über noetherschen Ringen (preprint).Google Scholar

Copyright information

© Springer-Verlag Wien 1984

Authors and Affiliations

  • Bruno J. Müller
    • 1
  1. 1.McMaster UniversityHamiltonCanada

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