Quasi-Frobenius Rings

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


A ring A is quasi-Frobenius (QF) in case A is right and left Artinian, and there exists an A-duality fin. gen. mod-A ↝ fin. gen. A-mod.


Hull Kato 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [39]
    Asano, K.: Über verallgemeinerte Abelsche Gruppen mit hyperkomplexen Operatorenring und ihre Anwendungen. Japan J. Math. Soc. Japan 15, 231–253 (1939).MathSciNetMATHGoogle Scholar
  2. [59]
    Azumaya, G.: A duality theory for injective modules (Theory of quasi-Frobenius modules). Amer. J. Math. 81, 249–278 (1959).MathSciNetCrossRefMATHGoogle Scholar
  3. [66]
    Azumaya, G.: Completely faithful modules and self-injective rings, Nagoya Math. J. 27, 697–708 (1966).MathSciNetMATHGoogle Scholar
  4. [43b]
    Baer, R.: Rings with duals. Amer. J. Math. 65, 569–584 (1943).MathSciNetCrossRefMATHGoogle Scholar
  5. [62a]
    Bass, H.: The Morita Theorems. Math. Dept., U. of Oreg., Eugene 1962.Google Scholar
  6. [62b]
    Bass, H.: Injective dimension in Noetherian rings. Trans. Amer. Math. Soc. 102, 18–29 (1962).MathSciNetCrossRefMATHGoogle Scholar
  7. [62c]
    Bass, H.: Torsion free and projective modules. Trans. Math. Soc. 102, 319–327 (1962).MathSciNetCrossRefMATHGoogle Scholar
  8. [63]
    Brauer, R.: Representations of finite groups. Lectures on Modern Mathematics, Vol. I, pp. 133–175 (T.L. Saaty, Ed.). John Wiley and Sons, Inc., New York 1953.Google Scholar
  9. [37]
    Brauer, R., Nesbitt, C.: On the regular representations of algebras. Proc. Nat. Acad. Sci. 23 (1937). Bridger, M. (see Auslander).Google Scholar
  10. [70a]
    Camillo, V. P.: Balanced rings and a problem of Thrall. Trans. Amer. Math. Soc. 149, 143–153 (1970).MathSciNetCrossRefMATHGoogle Scholar
  11. [72]
    Camillo, V. P., Fuller, K.R.: Balanced and QF-1 algebras. Proc. Amer. Math. Soc. 34, 373–378 (1972).MathSciNetMATHGoogle Scholar
  12. [66a]
    Cohn, P. M.: Morita Equivalence and Duality. University of London, Queen Mary College, Mile End Road, London, (Bookstore) 1966.Google Scholar
  13. [63]
    Connell, I.: On the group ring. Canad. J. Math. 15, 650–685 (1963).MathSciNetCrossRefMATHGoogle Scholar
  14. [73]
    Colby, R.R., Rutter, E.A., Jr.: Generalizations of QF-3 algebras. Trans. Amer. Math. Soc. 153, 371–385 (1973).MathSciNetGoogle Scholar
  15. [59]
    Curtis, C. W.: Quasi-Frobenius rings and Galois theory. Ill. J. Math. 3, 134–144 (1959).MathSciNetMATHGoogle Scholar
  16. [62]
    Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Interscience, New York 1962.Google Scholar
  17. [70]
    Dickson, S.E., Fuller, K.R.: Commutative QF-1 Artinian rings are QF. Proc. Amer. Math. Soc. 24, 667–670 (1970).MathSciNetMATHGoogle Scholar
  18. [58]
    Dieudonné, J.A.: Remarks on quasi-Frobenius rings. Ill. J. Math. 2, 346–354 (1958).MATHGoogle Scholar
  19. [72]
    Dlab and RingelGoogle Scholar
  20. [55]
    Eilenberg, S., Ikeda, M., Nakayama, T.: On the dimension of modules and algebras, I. Nagoya Math. J. 8, 49–57 (1955).MathSciNetMATHGoogle Scholar
  21. [56]
    Eilenberg, S., Nagao, H., Nakayama, T.: On the dimension of modules and algebras, IV. Dimension of residue rings of hereditary rings. Nagoya Math. J. 10, 87–95 (1956).MathSciNetMATHGoogle Scholar
  22. [55, 57]
    Eilenberg, S., Nakayama, T.: On the dimension of modules and algebras, II (Frobenius algebras and quasi-Frobenius rings). Nagoya Math. J. 9, 1–16 (1955);MathSciNetCrossRefMATHGoogle Scholar
  23. [55, 57]
    Eilenberg, S., Nakayama, T.: On the dimension of modules and algebras, IV (dimension of residue rings)loc. cit. 11, 9–12 (1957).MathSciNetMATHGoogle Scholar
  24. [69]
    Elizarov, V.P.: Quotient rings. Algebra and Logic 8, 219–243 (1969).MathSciNetCrossRefMATHGoogle Scholar
  25. [66a]
    Faith, C.: Rings with ascending condition on annihilators. Nagoya Math. J. 27, 179–191 (1966).MathSciNetMATHGoogle Scholar
  26. [76b]
    Faith, C.: Injective cogenerator rings, and Tachikawa’s theorem. Proc. Amer. Math. Soc. (1976).Google Scholar
  27. [76c]
    Faith, C.: Semiperfect Prüfer rings and FPF rings. Israel J. Math. 27, 113–119 (1976).MathSciNetMATHGoogle Scholar
  28. [76d]
    Faith, C.: Characterizations of rings by faithful modules. Lecture Notes, Math. Dept., Israel Institute of Technology (TECHNION), Haifa (1976).Google Scholar
  29. [67]
    Faith, C., Walker, E. A.: Direct sum representations of injective modules. J. Algebra 5, 203–221 (1967).MathSciNetCrossRefMATHGoogle Scholar
  30. [73]
    Farkas, D.: Self-injective group rings. J. Algebra 25, 313–315 (1973).MathSciNetCrossRefMATHGoogle Scholar
  31. [68]
    Floyd, D.R.: On QF-1 algebras. Pac. J. Math. 27, 81–94 (1968).MathSciNetMATHGoogle Scholar
  32. [70]
    Formanek, E.: A short proof of a theorem of Jennings. Proc. Amer. Math. Soc. 26, 406–407 (1970).MathSciNetCrossRefGoogle Scholar
  33. [68a]
    Fuller, K.R.: Generalized uniserial rings and their Kuppisch series. Math. Z. 106, 248–260 (1968).MathSciNetCrossRefMATHGoogle Scholar
  34. [68b]
    Fuller, K.R.: Structure of QF-3 rings. Trans. Amer. Math. Soc. 134, 343–354 (1968).MathSciNetMATHGoogle Scholar
  35. [69a]
    Fuller, K.R.: On indecomposable injectives over Artinian rings. Pac. J. Math. 29, 115–135 (1969).MathSciNetMATHGoogle Scholar
  36. [70a]
    Fuller, K.R.: Double centralizers of injectives and projectives over Artinian rings. Ill. J. Math. 14, 658–664 (1970).MathSciNetMATHGoogle Scholar
  37. [70b]
    Fuller, K.R.: Relative projectivity and injectivity classes determined by simple modules. J. Lond. Math. Soc. 5, 423–431 (1972).MathSciNetCrossRefMATHGoogle Scholar
  38. [70c]
    Fuller, K.R.: Primary rings and double centralizers. Pac. J. Math. 34, 379–383 (1970).MathSciNetMATHGoogle Scholar
  39. [67]
    Gewirtzman, L.: Anti-isomorphisms of the endomorphism rings of torsion-free modules. Math. Z. 98, 391–400 (1967).MathSciNetCrossRefMATHGoogle Scholar
  40. [73]
    Hannula, T.A.: On the construction of QF-rings. J. Algebra 25, 403–414 (1973).MathSciNetCrossRefMATHGoogle Scholar
  41. [66]
    Harada, M.: QF-3 and semiprimary PP-rings. Osaka J. Math. 2, 21–27 (1966).Google Scholar
  42. [64]
    Jacobson [64]Google Scholar
  43. [59a]
    Jans, J.P.: Projective injective modules. Pac. J. Math. 9, 1103–1108 (1959).MathSciNetMATHGoogle Scholar
  44. [59b]
    Jans, J.P.: On Frobenius algebras. Ann. of Math. 69, 392–407 (1959).MathSciNetCrossRefMATHGoogle Scholar
  45. [61]
    Jans, J.P.: Duality in Noetherian rings. Proc. Amer. Math. Soc. 12, 829–835 (1961).MathSciNetCrossRefMATHGoogle Scholar
  46. [67]
    Jans, J.P.: On orders in quasi-Frobenius rings. J. Algebra 7, 35–43 (1967).MathSciNetCrossRefMATHGoogle Scholar
  47. [70]
    Jans, J.P.: On the double centralizer property. Math. Ann. 188, 85–89 (1970).MathSciNetCrossRefGoogle Scholar
  48. [51]
    Ikeda, M.: Some generalizations of quasi-Frobenius rings. Osaka J. Math. 3, 227–239 (1951).MATHGoogle Scholar
  49. [52]
    Ikeda, M.: A characterization of quasi-Frobenius rings. Osaka J. Math. 4, 203–210 (1952).MATHGoogle Scholar
  50. [54]
    Ikeda, M., Nakayama, T.: On some characteristic properties of quasi-Frobenius and regular rings. Proc. Amer. Math. Soc. 5, 15–19 (1954).MathSciNetCrossRefMATHGoogle Scholar
  51. [41]
    Jennings, S. A.: The structure of the group ring of a p-group over a modular field. Trans. Amer. Math. Soc. 50, 175–185 (1941).MathSciNetGoogle Scholar
  52. [54]
    Kasch, F.: Grundlagen einer Theorie der Frobenius-Erweiterungen. Math. Ann. 127, 453–474 (1954).MathSciNetCrossRefMATHGoogle Scholar
  53. [60/61]
    Kasch, F.: Projektive Frobenius-Erweiterungen. Sitzungsber. Heidelberger Akad. 4, 89–109 (1960/61).Google Scholar
  54. [61]
    Kasch, F.: Dualitätseigenschaften von Frobenius-Erweiterungen. Math. Z. 77, 229–237 (1961).MathSciNetCrossRefGoogle Scholar
  55. [68a]
    Kato, T.: Some generalizations of QF-rings. Proc. Japan Acad. 44, 114–119 (1968).MathSciNetCrossRefMATHGoogle Scholar
  56. [68b]
    Kato, T.: Torsionless modules. Tohoku Math. J. 20, 234–243 (1968).MathSciNetCrossRefMATHGoogle Scholar
  57. [72]
    Kato, T.: Structure of left QF-3 rings. Proc. Japan Acad. 48, 479–483 (1972).MathSciNetCrossRefMATHGoogle Scholar
  58. [57]
    Kawada, Y.: On similarities and isomorphisms of ideals in a ring. J. Math. Soc. Japan 9, 374–380 (1957).MathSciNetCrossRefMATHGoogle Scholar
  59. [69]
    Klatt, G.B., Levy, L.S.: Pre-self-injective rings. Trans. Amer. Math. Soc. 122, 407–419 (1969).MathSciNetCrossRefGoogle Scholar
  60. [66a]
    Levy, L.S.: Commutative rings whose homomorphic images are self-injective. Pac. J. Math. 18, 149–153 (1966).MATHGoogle Scholar
  61. [71]
    Maisake [71]Google Scholar
  62. [69]
    Mewborn, A.C., Winton, C.N.: Orders in self-injective semiperfect rings. J. Algebra 13, 5–9 (1969).MathSciNetCrossRefMATHGoogle Scholar
  63. [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).MATHGoogle Scholar
  64. [62]
    Morita, K.: Category-isomorphisms and endomorphism rings of modules. Trans. Amer. Math. Soc. 103, 451–469 (1962).MathSciNetCrossRefMATHGoogle Scholar
  65. [66]
    Morita, K.: On S-rings in the sense of F.Kasch. Nagoya Math. J. 27, 688–695 (1966).Google Scholar
  66. [67]
    Morita, K.: The endomorphism ring theorem for Frobenius extensions. Math. Z. 102, 385–404 (1967).MathSciNetCrossRefMATHGoogle Scholar
  67. [69]
    Morita, K.: Duality in QF-3 rings. Math. Z. 108, 385–404 (1967).CrossRefGoogle Scholar
  68. [56]
    Morita, K., Tachikawa, H.: Character modules, submodules of a free module, and quasi-Frobenius rings. Math. Z. 65, 414–428 (1956).MathSciNetCrossRefMATHGoogle Scholar
  69. [64, 68]
    Müller, B. J.: Quasi-Frobenius-Erweiterungen I, II. Math. Z. 85, 345–368 (1964);MathSciNetCrossRefMATHGoogle Scholar
  70. [64, 68]a
    Müller, B. J.: Quasi-Frobenius-Erweiterungen I, II. Math. Z. 88, 380–409 (1968).CrossRefGoogle Scholar
  71. [39, 41]
    Nakayama, T.: On Frobeniusean algebras I, II. Ann. of Math. 40, 611–633 (1939);MathSciNetCrossRefGoogle Scholar
  72. [39, 41]a
    Nakayama, T.: On Frobeniusean algebras I, II. Ann. of Math. 42, 1–21 (1941).MathSciNetCrossRefGoogle Scholar
  73. [40a]
    Nakayama, T.: Note on uniserial and generalized uniserial rings. Proc. Imp. Acad. Tokyo 16, 285–289 (1940).MathSciNetCrossRefGoogle Scholar
  74. [40b]
    Nakayama, T.: Algebras with antiisomorphic left and right ideal lattices. Proc. Imp. Acad. Tokyo 17, 53–56 (1940).MathSciNetCrossRefGoogle Scholar
  75. [42]
    Nakayama, T.: On Frobeniusean algebras III. Japan J. Math. 18, 49–65 (1942).MathSciNetMATHGoogle Scholar
  76. [50a]
    Nakayama, T.: Supplementary remarks on Frobeniusean algebras II. Osaka Math. J. 2, 7–12 (1950).MathSciNetMATHGoogle Scholar
  77. [50b]
    Nakayama, T.: On two topics in the structural theory of rings (Galois theory and Frobenius algebras). Proc. ICM Vol. II, pp. 49–54 (1950).Google Scholar
  78. [71]
    Onodera, T.: Eine Bemerkung über Kogeneratoren. Proc. Japan Acad. 47, 140–141 (1971).MathSciNetCrossRefMATHGoogle Scholar
  79. [66]
    Osofsky [66]Google Scholar
  80. [71]
    Pareigis, B.: When Hopf algebras are Frobenius algebras. J. Algebra 18, 588–596 (1971).MathSciNetCrossRefMATHGoogle Scholar
  81. [71]
    Pareigis, B.: When Hopf algebras are Frobenius algebras. J. Algebra 18, 588–596 (1971).MathSciNetCrossRefMATHGoogle Scholar
  82. [70]
    Renault, G.: Sur les anneaux de groupes. Preprint. Faculté des Sciences, 86-Poitiers, Vienne 1970.Google Scholar
  83. [74]
    Ringel, C.M.: Commutative QF-1 rings. Proc. Amer. Math. Soc. 42, 365–368 (1974).MathSciNetMATHGoogle Scholar
  84. [75/6]
    Ringel and TachikawaGoogle Scholar
  85. [58]
    Tachikawa, H.: Duality theorem of character modules for rings with minimum condition. Math. Z. 68, 479–487 (1958).MathSciNetCrossRefMATHGoogle Scholar
  86. [62]
    Tachikawa, H.: A characterization of QF-3 algebras. Proc. Amer. Math. Soc. 13, 101–103 (1962).MathSciNetGoogle Scholar
  87. [69a]
    Tachikawa, H.: On splitting of module categories. Math. Z. 111, 145–150 (1969).MathSciNetCrossRefMATHGoogle Scholar
  88. [69b]
    Tachikawa, H.: A generalization of quasi-Frobenius rings. Proc. Amer. Math. Soc. 20, 471–476 (1969).MathSciNetCrossRefMATHGoogle Scholar
  89. [71]
    Tachikawa, H.: Localization and Artinian quotient rings. Math. Z. 119, 239–253 (1971).MathSciNetCrossRefMATHGoogle Scholar
  90. [73]
    Tachikawa, H.: Quasi-Frobenius Rings and Generalizations of QF-3 and QF-1 Rings. Lecture Notes in Mathematics. Springer, Berlin-Heidelberg-New York 1973.Google Scholar
  91. [56]
    Utumi, Y.: On quotient rings. Osaka Math. J. 8, 1–18 (1956).MathSciNetMATHGoogle Scholar
  92. [60a]
    Utumi, Y.: On continuous regular rings and semi-simple self-injective rings. Canad. J. Math. 12, 597–605 (1960).MathSciNetCrossRefMATHGoogle Scholar
  93. [60b]
    Utumi, Y.: A remark on quasi-Frobenius rings. Proc. Japan. Acad. 36, 15–17 (1960).MathSciNetCrossRefMATHGoogle Scholar
  94. [66]
    Utumi, Y.: On the continuity and self-injectivity of a complete regular ring. Canad. J. Math. 18, 404–412 (1966).MathSciNetCrossRefMATHGoogle Scholar
  95. [67]
    Utumi, Y.: Self-injective rings. J. Algebra 6, 56–64 (1967).MathSciNetCrossRefMATHGoogle Scholar
  96. [71]
    Wagoner [71]Google Scholar
  97. [67]
    Bass, H.: Lectures on topics in algebraic X-theory. Tata Institute for Advanced Study, Colaba 1967.Google Scholar
  98. [67]
    Singh, S., Jain, S.K.: On pseudo-injective modules and self-pseudo-injective rings. J. Math. Sci. India 2, 23–31 (1967).MathSciNetGoogle 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

Personalised recommendations