Skip to main content

Morita Duality

  • Chapter
Algebra II Ring Theory

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 191))

  • 797 Accesses

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Azumaya, G.: A duality theory for injective modules (Theory of quasi-Frobenius modules). Amer. J. Math. 81, 249–278 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  2. Baer, R.: Rings with duals. Amer. J. Math. 65, 569–584 (1943).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans. Amer. Math. Soc. 95, 466–488 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  4. Cohn, P.M.: On the embedding of rings in skew fields. Proc. Lond. Math. Soc. 11, 511–530 (1961).

    Article  MATH  Google Scholar 

  5. Cohn, P. M.: Morita Equivalence and Duality. University of London, Queen Mary College, Mile End Road, London, (Bookstore) 1966.

    Google Scholar 

  6. Curtis, C. W.: Quasi-Frobenius rings and Galois theory. Ill. J. Math. 3, 134–144 (1959).

    MathSciNet  MATH  Google Scholar 

  7. 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).

    MathSciNet  MATH  Google Scholar 

  8. Hochschild, G.: The structure of Lie groups. San Francisco, London, Amsterdam, Holden Day 1965.

    MATH  Google Scholar 

  9. 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. Ikeda, M.: Some generalizations of quasi-Frobenius rings. Osaka J. Math. 3, 227–239 (1951).

    MATH  Google Scholar 

  11. Ikeda, M.: A characterization of quasi-Frobenius rings. Osaka J. Math. 4, 203–210 (1952).

    MATH  Google Scholar 

  12. Ikeda, M., Nakayama, T.: On some characteristic properties of quasi-Frobenius and regular rings. Proc. Amer. Math. Soc. 5, 15–19 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  13. Jans, J.P.: Duality in Noetherian rings. Proc. Amer. Math. Soc. 12, 829–835 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  14. Jategaonkar, A. V.: Jacobson’s conjecture and modules over fully bounded noetherian rings. J. Algebra 30, 103–121 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  15. Jategaonkar, A. V.: Injective modules and localization in non-commutative noetherian rings. Trans. Amer. Soc. 190, 109–123 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  16. Kaplansky, I.: Dual modules over a valuation ring. Proc. Amer. Math. Soc. 4, 213–219 (1953).

    Article  MathSciNet  MATH  Google Scholar 

  17. Lambek, J., Rattray, B.: Localizations and duality in additive categories. Houston J. Math. 1, 87–100 (1975).

    MathSciNet  MATH  Google Scholar 

  18. Matlis, E.: Injective modules over noetherian rings. Pac. J. Math. 8, 511:528 (1958).

    MathSciNet  Google Scholar 

  19. 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).

    MATH  Google Scholar 

  20. Morita, K.: The endomorphism ring theorem for Frobenius extensions. Math. Z. 102, 385–404 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  21. Morita, K.: Duality in QF-3 rings. Math. Z. 108, 385–404 (1967).

    Article  Google Scholar 

  22. Morita, K., Kawada, Y, Tachikawa, H.: On injective modules. Math. Z. 68, 217–218 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  23. Müller, B.J.: Linear compactness and Morita duality. J. Algebra 16, 60–66 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  24. Müller, B. J.: Duality theory for linearly topologized modules. Math. Z. 119, 63–74 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  25. Nagoa and Nakayama

    Google Scholar 

  26. Nakayama, T.: On Frobeniusean algebras I. Ann. of Math. 40, 611–633 (1939).

    Article  MathSciNet  Google Scholar 

  27. Nakayama, T.: On Frobeniusean algebras II. Ann. of Math. 40, 42, 1–21 (1941).

    Article  MathSciNet  Google Scholar 

  28. Nakayama, T.: Note on uniserial and generalized uniserial rings. Proc. Imp. Acad. Tokyo 16, 285–289 (1940).

    Article  MathSciNet  Google Scholar 

  29. Nakayama, T.: Algebras with antiisomorphic left and right ideal lattices. Proc. Imp. Acad. Tokyo 17, 53–56 (1940).

    Article  MathSciNet  Google Scholar 

  30. Oberst, U.: Duality theory for Grothendieck categories and linearly compact rings. J. Algebra 15, 473–542(1970).

    Article  MathSciNet  MATH  Google Scholar 

  31. Onodera, T.: Linearly compact modules and cogenerators. J. Fac. Sci. Hokkaido U. Ser. I 22, 116–125(1972).

    MathSciNet  MATH  Google Scholar 

  32. Onodera, T.: Linearly compact modules and cogenerators II. Hokkaido Math. J. 2, 243–251 (1973).

    MathSciNet  MATH  Google Scholar 

  33. Ornstein

    Google Scholar 

  34. Osofsky, B.L.: Cyclic injective modules of full linear rings. Proc. Amer. Math. Soc. 17 247–253 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  35. Osofsky, B. L.: A generalization of quasi-Frobenius rings. J. Algebra 4, 373–387 (1966);

    Article  MathSciNet  MATH  Google Scholar 

  36. Osofsky, B. L.: A generalization of quasi-Frobenius rings. Erratum 9, 120 (1968).

    MathSciNet  Google Scholar 

  37. Osofsky, B.L.: Erratum. J. Algebra 9, 120 (1968) (see Osofsky [66b]).

    Article  MathSciNet  Google Scholar 

  38. Pontryagin, L.: Topological Groups. Princeton U. Press, Princeton 1939.

    MATH  Google Scholar 

  39. Rosenberg, A.: Blocks and centres of group algebras. Math. Z. 76, 209–216 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  40. Sandomierski, F. L.: Modules over the endomorphism rings of a finitely generated projective module. Proc. Amer. Math. Soc. 31, 27–31 (1971).

    Article  MathSciNet  Google Scholar 

  41. Sandomierski, F.L.: Linearly compact modules and local Morita duality. Ring Theory.

    Google Scholar 

  42. Tachikawa, H.: Duality theorem of character modules for rings with minimum condition. Math. Z. 68, 479–487 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  43. Tachikawa, H.: On rings for which every indecomposable right module has a unique maximal submodule. Math. Z. 71, 200–222 (1959).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1976 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Faith, C. (1976). Morita Duality. In: Faith, C. (eds) Algebra II Ring Theory. Grundlehren der mathematischen Wissenschaften, vol 191. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65321-6_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-65321-6_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-65323-0

  • Online ISBN: 978-3-642-65321-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics