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Morita Duality

  • Carl Faith
Chapter
  • 554 Downloads
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 191)

Abstract

In this chapter we introduce Morita duality. Roughly speaking, these theorems are dual to the Morita theorems on category equivalence (Chapter 12).

Keywords

Abelian Category Artinian Ring Contravariant Functor Finite Dimensional Algebra Injective Hull 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [59]
    Azumaya, G.: A duality theory for injective modules (Theory of quasi-Frobenius modules). Amer. J. Math. 81, 249–278 (1959).MathSciNetzbMATHCrossRefGoogle Scholar
  2. [43b]
    Baer, R.: Rings with duals. Amer. J. Math. 65, 569–584 (1943).MathSciNetzbMATHCrossRefGoogle Scholar
  3. [60]
    Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans. Amer. Math. Soc. 95, 466–488 (1960).MathSciNetzbMATHCrossRefGoogle Scholar
  4. [61]
    Cohn, P.M.: On the embedding of rings in skew fields. Proc. Lond. Math. Soc. 11, 511–530 (1961).zbMATHCrossRefGoogle Scholar
  5. [66]
    Cohn, P. M.: Morita Equivalence and Duality. University of London, Queen Mary College, Mile End Road, London, (Bookstore) 1966.Google Scholar
  6. [59]
    Curtis, C. W.: Quasi-Frobenius rings and Galois theory. Ill. J. Math. 3, 134–144 (1959).MathSciNetzbMATHGoogle Scholar
  7. [69]
    Dickson, S. E., Fuller, K.R.: Algebras for which every indecomposable right module is invariant in its injective envelope. Pac. J. Math. 31, 655–658 (1969).MathSciNetzbMATHGoogle Scholar
  8. [65]
    Hochschild, G.: The structure of Lie groups. San Francisco, London, Amsterdam, Holden Day 1965.zbMATHGoogle Scholar
  9. [70]
    Hoffman, K.H.: The duality of compact semigroups and C*-bigebras. Lecture Notes in Mathematics, vol. 129, Springer, Berlin-Heidelberg-New York 1970.Google Scholar
  10. [51]
    Ikeda, M.: Some generalizations of quasi-Frobenius rings. Osaka J. Math. 3, 227–239 (1951).zbMATHGoogle Scholar
  11. [52]
    Ikeda, M.: A characterization of quasi-Frobenius rings. Osaka J. Math. 4, 203–210 (1952).zbMATHGoogle Scholar
  12. [54]
    Ikeda, M., Nakayama, T.: On some characteristic properties of quasi-Frobenius and regular rings. Proc. Amer. Math. Soc. 5, 15–19 (1954).MathSciNetzbMATHCrossRefGoogle Scholar
  13. [61]
    Jans, J.P.: Duality in Noetherian rings. Proc. Amer. Math. Soc. 12, 829–835 (1961).MathSciNetzbMATHCrossRefGoogle Scholar
  14. [74a]
    Jategaonkar, A. V.: Jacobson’s conjecture and modules over fully bounded noetherian rings. J. Algebra 30, 103–121 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  15. [74b]
    Jategaonkar, A. V.: Injective modules and localization in non-commutative noetherian rings. Trans. Amer. Soc. 190, 109–123 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  16. [53]
    Kaplansky, I.: Dual modules over a valuation ring. Proc. Amer. Math. Soc. 4, 213–219 (1953).MathSciNetzbMATHCrossRefGoogle Scholar
  17. [75]
    Lambek, J., Rattray, B.: Localizations and duality in additive categories. Houston J. Math. 1, 87–100 (1975).MathSciNetzbMATHGoogle Scholar
  18. [58]
    Matlis, E.: Injective modules over noetherian rings. Pac. J. Math. 8, 511:528 (1958).MathSciNetGoogle Scholar
  19. [58]
    Morita, K.: Duality for modules and its applications to the theory of rings with minimum condition. Sci Rpts. Tokyo Kyoiku Daigaku 6, 83–142 (1958).zbMATHGoogle Scholar
  20. [67]
    Morita, K.: The endomorphism ring theorem for Frobenius extensions. Math. Z. 102, 385–404 (1967).MathSciNetzbMATHCrossRefGoogle Scholar
  21. [69]
    Morita, K.: Duality in QF-3 rings. Math. Z. 108, 385–404 (1967).CrossRefGoogle Scholar
  22. [57]
    Morita, K., Kawada, Y, Tachikawa, H.: On injective modules. Math. Z. 68, 217–218 (1957).MathSciNetzbMATHCrossRefGoogle Scholar
  23. [70b]
    Müller, B.J.: Linear compactness and Morita duality. J. Algebra 16, 60–66 (1970).MathSciNetzbMATHCrossRefGoogle Scholar
  24. [71]
    Müller, B. J.: Duality theory for linearly topologized modules. Math. Z. 119, 63–74 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  25. [53]
    Nagoa and NakayamaGoogle Scholar
  26. [39]
    Nakayama, T.: On Frobeniusean algebras I. Ann. of Math. 40, 611–633 (1939).MathSciNetCrossRefGoogle Scholar
  27. [41]
    Nakayama, T.: On Frobeniusean algebras II. Ann. of Math. 40, 42, 1–21 (1941).MathSciNetCrossRefGoogle Scholar
  28. [40a]
    Nakayama, T.: Note on uniserial and generalized uniserial rings. Proc. Imp. Acad. Tokyo 16, 285–289 (1940).MathSciNetCrossRefGoogle Scholar
  29. [40b]
    Nakayama, T.: Algebras with antiisomorphic left and right ideal lattices. Proc. Imp. Acad. Tokyo 17, 53–56 (1940).MathSciNetCrossRefGoogle Scholar
  30. [70]
    Oberst, U.: Duality theory for Grothendieck categories and linearly compact rings. J. Algebra 15, 473–542(1970).MathSciNetzbMATHCrossRefGoogle Scholar
  31. [72]
    Onodera, T.: Linearly compact modules and cogenerators. J. Fac. Sci. Hokkaido U. Ser. I 22, 116–125(1972).MathSciNetzbMATHGoogle Scholar
  32. [73a]
    Onodera, T.: Linearly compact modules and cogenerators II. Hokkaido Math. J. 2, 243–251 (1973).MathSciNetzbMATHGoogle Scholar
  33. [66]
  34. [66a]
    Osofsky, B.L.: Cyclic injective modules of full linear rings. Proc. Amer. Math. Soc. 17 247–253 (1966).MathSciNetzbMATHCrossRefGoogle Scholar
  35. [66b]
    Osofsky, B. L.: A generalization of quasi-Frobenius rings. J. Algebra 4, 373–387 (1966);MathSciNetzbMATHCrossRefGoogle Scholar
  36. [66c]
    Osofsky, B. L.: A generalization of quasi-Frobenius rings. Erratum 9, 120 (1968).MathSciNetGoogle Scholar
  37. [68 e]
    Osofsky, B.L.: Erratum. J. Algebra 9, 120 (1968) (see Osofsky [66b]).MathSciNetCrossRefGoogle Scholar
  38. [39]
    Pontryagin, L.: Topological Groups. Princeton U. Press, Princeton 1939.zbMATHGoogle Scholar
  39. [61]
    Rosenberg, A.: Blocks and centres of group algebras. Math. Z. 76, 209–216 (1961).MathSciNetzbMATHCrossRefGoogle Scholar
  40. [72a]
    Sandomierski, F. L.: Modules over the endomorphism rings of a finitely generated projective module. Proc. Amer. Math. Soc. 31, 27–31 (1971).MathSciNetCrossRefGoogle Scholar
  41. [72b]
    Sandomierski, F.L.: Linearly compact modules and local Morita duality. Ring Theory.Google Scholar
  42. [58]
    Tachikawa, H.: Duality theorem of character modules for rings with minimum condition. Math. Z. 68, 479–487 (1958).MathSciNetzbMATHCrossRefGoogle Scholar
  43. [59]
    Tachikawa, H.: On rings for which every indecomposable right module has a unique maximal submodule. Math. Z. 71, 200–222 (1959).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • Carl Faith
    • 1
    • 2
  1. 1.Rutgers, The State UniversityNew BrunswickUSA
  2. 2.The Institute for Advanced StudyPrincetonUSA

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