Skip to main content

Frobenius algebras and quasi-Frobenius rings

  • Chapter
  • 1924 Accesses

Part of the book series: Mathematics and Its Applications ((MAIA,volume 586))

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
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
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. I. Assen, J. Nehring, A. Skowroúski, Algebras with cyclic-finite derived categories, Math. Ann., v.280, 1988, p.441-463.

    Article  MathSciNet  Google Scholar 

  2. M. Auslander, On the dimension of modules and algebras (III) (Global dimension), Nagoya Math. J., v.9, 1955, p.67-77.

    MATH  MathSciNet  Google Scholar 

  3. R. Brauer, C. Nesbitt, On the regular representations of algebras, Proc. Nat. Acad. Sci., v.23, 1937, p.236-240.

    Article  MATH  Google Scholar 

  4. O. Bretscher, C. Läser, C. Riedtmann, Selfinjective and simply connected algebras, Manuscripta Math., v.36, 1982, p.253-307.

    Article  Google Scholar 

  5. C. Curtis, J. Reiner, Representation theory of finite groups and associative algebras, John Wiley and Sons, 1962.

    Google Scholar 

  6. J. Dieudonné, Remarks on quasi-Frobenius rings, Illinois J. Math., v.2, 1958, p.346-354.

    MATH  MathSciNet  Google Scholar 

  7. H. Dinh, S.R. Lopez-Permouth, On the equivalence of codes over finite rings, Appl. Algebra Engrg. Comm. Comput., v.15, 2004, N.1, p.37-50.

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Dinh, S.R. Lopez-Permouth, On the equivalence of codes over rings and modules, Finite Fields Appl., v.10, 2004, N.4, p. 615-625.

    Article  MATH  MathSciNet  Google Scholar 

  9. M.A. Dokuchaev, V.V. Kirichenko, Quasi-Frobenius rings and Nakayama permutations of semi-perfect rings, Ukr. Math. J., v. 54, N 7, 2002, p. 919-930.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Eilenberg, T. Nakayama, On dimensions of modules and algebras, II (Frobenius algebras and quasi-Frobenius rings), Nagoya Math. J., v.9, 1955, p.1-16.

    MATH  MathSciNet  Google Scholar 

  11. K. Erdmann, Blocks of Tame Representation Type and Related Algebras, Springer Lecture Notes in Math., v.1428, Springer, Berlin, 1990.

    Google Scholar 

  12. K. Erdmann, A. Skowroński, On Auslander-Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc., v.330, 1992, p.169-189.

    Article  Google Scholar 

  13. C. Faith, E.A. Walker, Direct-sum representations of injective modules, J. Algebra, v.5, 1967, p.203-221.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Frobenius, Theorie der hyperkomplexen Gröβen I,II, Berlin Ber., 1903, p.504-538, p.634-645.

    Google Scholar 

  15. K.R. Goodearl, Simple self-injective rings need not be Artinian, Comm. Algebra, v.2, 1974, p.83-89.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Gordon, L.W. Small, Piecewise domains, J. Algebra, v. 23, 1972, p.553-564.

    Article  MATH  MathSciNet  Google Scholar 

  17. Edward L. Green, Frobenius algebras and their quivers, Can. J. Math., v.30, N 5, 1978, p. 1029-1044.

    MATH  Google Scholar 

  18. M. Greferath, A. Nechaev, R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl., v.3, No.3, 2004, p.247-272.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Greferath, M.E. O’Sullivan, On bounds for codes over Frobenius rings under homogeneous weights, Discrete Math., v.289, 2004, p.1-3, p.11-24.

    Article  MathSciNet  Google Scholar 

  20. T.A. Hannula, On the construction of Quasi-Frobenius Rings, J. Algebra, v.25, 1973, p.403-414.

    Article  MATH  MathSciNet  Google Scholar 

  21. D. Hughes, J. Waschbüsch, Trivial extensions of tilted algebras, Proc. London Math. Soc., v.46, 1983, p.347-364.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Ikeda, A characterization of Quasi-Frobenius Rings, Osaka Math. J., v.4, No.2, 1952, p.203-209.

    MATH  MathSciNet  Google Scholar 

  23. M. Ikeda, T. Nakayama, On some characteristic properties of Quasi-Frobenius and regular rings, Proc. Amer. Math. Soc., v.5, 1954, p.15-19.

    Article  MATH  MathSciNet  Google Scholar 

  24. H. Kasch, Modules and Rings, London Mathematical Society Monographs, v.17, Academic Press, 1982.

    Google Scholar 

  25. V.V. Kirichenko, Generalized uniserial rings, Mat. Sbornik, v.99 (141), N4, 1976 (in Russian); English translation: Math. USSR Sbornik, v.28, N4, 1976, pp. 501-520.

    Google Scholar 

  26. V.V. Kirichenko, Samir Valio, Yu. V. Yaremenko, Semiperfect rings and their quivers, Infinite Groups and Related Topics, Kiev, Inst. Math. NAS Ukraine, 1993, p.438-456 (in Russian).

    Google Scholar 

  27. H. Kupisch, Symmetrische Algebren mit endlich vielen unzerlegbaren Darstellungen, I, J. Reine Angew. Math., v.219, 1965, p.1-25.

    MATH  MathSciNet  Google Scholar 

  28. H. Kupisch, Symmetrische Algebren mit endlich vielen unzerlegbaren Darstellungen, II, J. Reine Angew. Math., v.245, 1970, p.1-13.

    MATH  MathSciNet  Google Scholar 

  29. H. Kupisch, Quasi-Frobenius algebras of finite representation type, Springer Lecture Notes in Math., v.488, Springer, Berlin, 1975, p.184-200.

    Article  MathSciNet  Google Scholar 

  30. K. Morita, H. Tachikawa, Character modules, submodules of a free module, and quasi-Frobenius rings, Math. Z., v.65, 1956, p.414-428.

    Article  MATH  MathSciNet  Google Scholar 

  31. K. Morita, Duality for modules and its applications of the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiki Daigaku Soc., A6, 1958, p.83-142.

    Google Scholar 

  32. K. Morita, Duality in QF-3 rings, Math. Z., v.108, 1969, p.237-252.

    Article  MATH  MathSciNet  Google Scholar 

  33. B. Müller, The structure of quasi-Frobenius rings, Can. J. Math., v.26, 1974, p.1141-1151.

    MATH  Google Scholar 

  34. T. Nakayama, On Frobenius algebra, I, Ann. Math., v.40, 1939, p.611-633.

    Article  MathSciNet  Google Scholar 

  35. T. Nakayama, On Frobeniusean algebras, II, Ann. Math., v.42(1), 1941, p.1-21.

    Article  MathSciNet  Google Scholar 

  36. T. Nakayama, M. Ikeda, Supplementary Remarks on Frobeniusean algebras II, Osaka Math. J., v.2, No.1, 1950, p.7-12.

    MATH  MathSciNet  Google Scholar 

  37. J. Nehring, Polynomial growth trivial extensions of non-simply connected algebras, Bull. Polish Acad. Sci., v.9/10, 1989.

    Google Scholar 

  38. J. Nehring, A. Skowroński, Polynomial growth trivial extensions of simply connected algebras, Fund. Math., v.132, 1989, p.117-134.

    MATH  MathSciNet  Google Scholar 

  39. C. Nesbitt, On the regular representations of algebras, Ann. Math., v.39, 1938, p.634-658.

    Article  MathSciNet  Google Scholar 

  40. D.G. Northcott, A first course of homological algebra, Cambridge University Press, 1973.

    Google Scholar 

  41. K. Oshiro, S. Rim, On QF-rings with cyclic Nakayama permutations, Osaka J. Math., v.34, 1997, p.1-19.

    MATH  MathSciNet  Google Scholar 

  42. B.L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra, v.4, 1966, p. 373-387.

    Article  MathSciNet  Google Scholar 

  43. B.L. Osofsky, Cyclic injective modules of full linear rings, Proc. Amer. Math. Soc., v.17, 1966, p.247-253.

    Article  MATH  MathSciNet  Google Scholar 

  44. B. Osofsky, A semiperfect one-sided injective ring, Communications in Algebra, v.12(16), 1984, p.2037-2041.

    Article  MATH  MathSciNet  Google Scholar 

  45. C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungegen und zurück, Comment. Math. Helv., v.55, 1980, p.199-224.

    Article  MATH  Google Scholar 

  46. C. Riedtmann, Representation-finite selfinjective algebras of type A_n, In: Representation Theory II, Lecture Notes in Math., v.831, Springer-Verlag, Berlin, 1980, p.449-520.

    Google Scholar 

  47. C. Riedtmann, Representation-finite selfinjective algebras of type D_n, Compositio Math., v.49, 1983, p.231-282.

    MATH  MathSciNet  Google Scholar 

  48. F.L. Sandomerski, Some examples of right self-injective rings which are not left self-injective, Proc. Amer. Math. Soc., v.26, 1970, p.244-245.

    Article  MathSciNet  Google Scholar 

  49. A. Skowroński, Selfinjective algebras of polynomial growth, Math. Ann., v.285, 1989, p.177-199.

    Article  MATH  MathSciNet  Google Scholar 

  50. R.M. Thrall, Some generalizations of quasi-Frobenius algebras, Trans. Amer. Math.Soc., v.64, 1948, p.173-183.

    Article  MATH  MathSciNet  Google Scholar 

  51. J. Waschbüsch, Symmetrische Algebren vom endlichen Modultyp, J. Reine Angew. Math., v.321, 1981, p.78-98.

    MATH  MathSciNet  Google Scholar 

  52. J.A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., v.121, N.3, 1999, p.555-575.

    Article  MATH  MathSciNet  Google Scholar 

  53. M. Yousif, Rings and Nakayama permutations, Communications in Algebra, v.25(12), 1997, p.3787-3795.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer

About this chapter

Cite this chapter

Hazewinkel, M., Gubareni, N., Kirichenko, V. (2007). Frobenius algebras and quasi-Frobenius rings. In: Hazewinkel, M., Gubareni, N., Kirichenko, V. (eds) Algebras, Rings and Modules. Mathematics and Its Applications, vol 586. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5141-8_4

Download citation

Publish with us

Policies and ethics