Projective Covers and Perfect Rings

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


A morphism f: AB of R-modules is said to be minimal provided that ker f is a superfluous submodule of A. For example, for a right ideal I, the canonical map R → R/I is superfluous if and only if I ⊆ rad R 18.3. A module A is a projective cover (proj. cov.) of B provided that A is projective and there exists a minimal epimorphism A → B. This notion is dual to that of injective hull, and yet, although each R-module has an injective hull, projective covers of modules may fail to exist. For example, as is shown in this chapter, a necessary condition that every right R-module have a projective cover is that R/rad R be semisimple, and rad R be a nil ideal.


Local Ring Left Ideal Projective Module Ring Theory Projective Cover 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [60]
    Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans. Amer. Math. Soc. 95, 466–488 (1960).MathSciNetzbMATHCrossRefGoogle Scholar
  2. [69]
    Björk, J.E.: Rings satisfying a minimum condition on principal ideals. J. reine u. angew. Math. 236, 466–488 (1969).Google Scholar
  3. [70a]
    Björk, J.E.: On subrings of matrix rings over fields. Proc. Cam. Phil. Soc. 68, 275–284 (1970). Brauer and Weiss [64]zbMATHCrossRefGoogle Scholar
  4. [74]
    Camillo, V. P., Fuller, K.R.: On Loewy length of rings. Pac. J. Math. 53, 347–354 (1974).MathSciNetzbMATHGoogle Scholar
  5. [60]
    Chase, S.U.: Direct products of modules. Trans. Amer. Math. Soc. 97, 457–473 (1960).MathSciNetCrossRefGoogle Scholar
  6. [63]
    Connell, I.: On the group ring. Canad. J. Math. 15, 650–685 (1963).MathSciNetzbMATHCrossRefGoogle Scholar
  7. [69]
    Courter, R.C.: Finite direct sums of complete matrix rings over perfect completely primary rings. Canad. J. Math. 21, 430–446 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
  8. [70]
    Cozzens, J. H.: Homological properties of the ring of differential polynomials. Bull. Amer. Math. Soc. 76, 75–79 (1970).MathSciNetzbMATHCrossRefGoogle Scholar
  9. [70]
    Dickson, S.E., Fuller, K.R.: Commutative QF-1 Artinian rings are QF. Proc. Amer. Math. Soc. 24, 667–670 (1970).MathSciNetzbMATHGoogle Scholar
  10. [70]
  11. [56]
    Eilenberg, S.: Homological dimension and syzygies. Ann. of Math. 64, 328–336 (1956). ilenberg, S. (see Cartan).MathSciNetzbMATHCrossRefGoogle Scholar
  12. [70a]
    Fuchs, L.: Abelian Groups (Second Edition), Vol.1. Pergamon, New York 1970.Google Scholar
  13. [70b]
    Fuchs, L.: Torsion preradicals and ascending Loewy series of modules. J. reine u. angew. Math. 239, 169–179 (1970).Google Scholar
  14. [69b]
    Fuller, K.R.: On direct representations of quasi-injectives and quasi-projectives. Arch. Math. 20, 495–502 (1969);MathSciNetzbMATHCrossRefGoogle Scholar
  15. [69c]
    Fuller, K.R.: On direct representations of quasi-injectives and quasi-projectives. Arch. Math. 21, 478 (1970).MathSciNetzbMATHCrossRefGoogle Scholar
  16. [70a]
    Fuller, K.R.: Double centralizers of injectives and projectives over Artinian rings. Ill. J. Math. 14, 658–664 (1970).MathSciNetzbMATHGoogle Scholar
  17. [72]
    Golan, J. S.: Characterization of rings using quasiprojective modules, III. Proc. Amer. Math. Soc. 31 (1972).Google Scholar
  18. [68]
    Gupta, R.N.: Characterization of rings whose classical quotient rings are perfect rings.Google Scholar
  19. [62]
  20. [70]
    Jonah, D.: Rings with minimum condition for principal right ideals have the maximum condition for principal left ideals. Math. Z. 113, 106–112 (1970).MathSciNetzbMATHCrossRefGoogle Scholar
  21. [66]
    Kasch, F., Mares, E.A.: Eine Kennzeichnung semi-perfekter Moduln. Nagoya Math. J. 27, 525–529 (1966).MathSciNetzbMATHGoogle Scholar
  22. [70]
    Koehler, A.: Quasi-projective covers and direct sums. Proc. Amer. Math. Soc. 24, 655–658 (1970).MathSciNetzbMATHCrossRefGoogle Scholar
  23. [30a]
    Köthe, G.: Die Struktur der Ringe deren Restklassenring nach dem Radical vollständig reduzibel ist. Math. Z. 32, 161–186 (1930).MathSciNetzbMATHCrossRefGoogle Scholar
  24. [28]
    Krull, W.: Zur Theorie der Allgemeinen Zahlringe. Math. Ann. 99, 51–70 (1928).MathSciNetzbMATHCrossRefGoogle Scholar
  25. [05]
    Loewy, A.: Über die vollständig reduciblen Gruppen, die zu einer Gruppe linearer homogener Substitutionen gehören. Trans. Amer. Math. Soc. 6, 504–533 (1905).MathSciNetzbMATHGoogle Scholar
  26. [17]
    Loewy, A.: Über Matrizen und Differentialkomplexe, I. Math. Ann. 78, 1–51, 343–368 (1917).MathSciNetCrossRefGoogle Scholar
  27. [17]
    Loewy, A.: Über Matrizen und Differentialkomplexe, II. Math. Ann. 78, 1–51, 343–368 (1917).MathSciNetCrossRefGoogle Scholar
  28. [63]
    Mares, E. A.: Semiperfect modules. Math. Z. 82, 347–360 (1963).MathSciNetzbMATHCrossRefGoogle Scholar
  29. [69a]
    Michler, G.O.: On quasi-local noetherian rings. Proc. Amer. Math. Soc. 20, 222–224 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
  30. [69b]
    Michler, G.O.: Structure of semiperfect hereditary Noetherian rings. J. Algebra 13, 327–344 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
  31. [69c]
    Michler, G.O.: Idempotent ideals in perfect rings. Canad. J. Math. 21, 301–309 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
  32. [69d]
    Michler, G.O.: Asano orders. Proc. Lond. Math. Soc. 19, 421–443 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
  33. [70a]
    Müller, B. J.: On semiperfect rings. Ill. J. Math. 14, 464–467 (1970).zbMATHGoogle Scholar
  34. [70b]
    Müller, B.J.: Linear compactness and Morita duality. J. Algebra 16, 60–66 (1970).MathSciNetzbMATHCrossRefGoogle Scholar
  35. [71b]
    Osofsky, B.L.: Loewy length of perfect rings. Proc. Amer. Math. Soc. 28, 352–354 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  36. [72]
  37. [70]
    Cailleau, A, Renault, G.: Etude des modules ∑-injective. C. R. Acad. Sci. Paris 270, 1391–1394 (1970).zbMATHGoogle Scholar
  38. [67]
    RentschlerGoogle Scholar
  39. [61]
    Rosenberg, A.: Blocks and centres of group algebras. Math. Z. 76, 209–216 (1961).MathSciNetzbMATHCrossRefGoogle Scholar
  40. [64]
    Sandomierski, F.: Relative injectivity and projectivity. Ph. D. Thesis. Penna, State U., U. Park 1964.Google Scholar
  41. [71]
    Shores, T. S.: Decompositions of finitely generated modules. Proc. Amer. Math. Soc. 30, 445–450 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  42. [74]
    Shores, T.S.: Loewy series of modules. J. reine u. angew. Math. 265, 183–200 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  43. [60]
    Swan, R.: Induced representations and projective modules. Ann. of Math. 71, 552–578 (1960).MathSciNetzbMATHCrossRefGoogle Scholar
  44. [68]
    Swan, R.G.: Algebraic K-Theory. Lecture Notes in Mathematics, vol.76. Springer, Berlin-Heidelberg-New York 1968.Google Scholar
  45. [70]
    Teply, M.L.: Homological dimension and splitting torsion theories. Pac. J. Math. 34, 193–205 (1970).MathSciNetzbMATHGoogle Scholar
  46. [72a]
    Warfield, R.B., Jr.: Rings whose modules have nice decompositions. Math. Z. 125, 187–192 (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  47. [72b]
    Warfield, R.B., Jr.: Exchange rings and decompositions of modules. Math. Ann. 199, 31–36 (1972).MathSciNetzbMATHCrossRefGoogle Scholar
  48. [67]
    Jans, J., Wu, L.: On quasi-projectives. Ill. J. Math. 11, 439–448 (1967).MathSciNetzbMATHGoogle Scholar
  49. [70]
  50. [73]
    Zöschinger, H.: Moduln, die in jeder Erweiterung ein Komplement haben. Algebra-Berichte-Seminar Kasch und Pareigis. Math. Inst. München 15, 1–22 (1973).Google Scholar
  51. [61]
    Albrecht, F.: On projective modules over semi-hereditary rings. Proc. Amer. Math. Soc. 12, 638–639 (1861).MathSciNetCrossRefGoogle Scholar
  52. Azumaya [75]Google Scholar
  53. [58a]
    Kaplansky, I.: Projective modules. Ann. of Math. 68, 372–377 (1958).MathSciNetzbMATHCrossRefGoogle Scholar
  54. [26]
    Krull, W.: Theorie und Anwendung der verallgemeinerten Abelschen Gruppen. Sitzungsber. Heidelberger Akad. 7, 1–32 (1926).Google 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