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Algebra pp 365-388 | Cite as

Semisimple Modules and Homological Dimension

  • Carl Faith
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 190)

Abstract

A module M R is semisimple in case M is a direct sum Ц i I M i of a family of simple submodules {M i | iI}. A ring R is semisimple in case R R is. We have seen that any simple Artinian ring is semisimple. Finite ring products of simple Artinian rings are also semisimple. The Wedderburn-Artin theorem implies the converse: Every semisimple ring is isomorphic to a finite product of full matrix rings of various degrees over various fields. These rings are also characterized by the Wedderburn-Artin theorem (8.8) as right Artinian rings with no nilpotent ideals. (The condition is right-left symmetric.)

Keywords

Global Dimension Left Ideal Projective Dimension Quotient Ring Semiprime Ring 
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. [61]
    Albrecht, F.: On projective modules over semi-hereditary rings. Proc. Amer. Math. Soc. 12, 638–639 (1961).CrossRefzbMATHMathSciNetGoogle Scholar
  2. [27a]
    Artin, E.: Zur Theorie der hyperkomplexen Zahlen. Abh. Math. Sem. Univ. Hamburg 5, 251–260 (1927).CrossRefzbMATHGoogle Scholar
  3. [27b]
    Artin, E.: Zur Arithmetik hyperkomplexer Zahlen. Abh. Math. Sem. Univ. Hamburg 5, 261–289 (1927).Google Scholar
  4. [55]
    Auslander, M.: On the dimension of modules and algebras, III. Nagoya Math. J. 9, 67–77 (1955).zbMATHMathSciNetGoogle Scholar
  5. [57]
    Auslander, M., Buchsbaum, D.: Homological dimension in local rings. Trans. Amer. Math. Soc. 85, 390–405 (1957).CrossRefzbMATHMathSciNetGoogle Scholar
  6. [60]
    Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc. 95, 466–488 (1960).CrossRefzbMATHMathSciNetGoogle Scholar
  7. [56]
    Cartan, H., Eilenberg, S.: Homological Algebra. Princeton: Princeton University Press 1956.zbMATHGoogle Scholar
  8. [71]
    Eisenbud, D., Griffith, P.: Serial rings. J. Algebra 17, 389–400 (1971).CrossRefzbMATHMathSciNetGoogle Scholar
  9. [66a]
    Faith, C.: Rings with ascending condition on annihilators. Nagoya Math. J.27, 179–191 (1966).zbMATHMathSciNetGoogle Scholar
  10. [72a]
    Faith, C.: A correspondence theorem for projective modules, and the structure of simple Noetherian rings. Proceedings of the Conference on Associative Algebras, Nov. 1970. Istituto Nazionale di Alta Matematica, Symposium Matematica 8, 309–345 (1972).MathSciNetGoogle Scholar
  11. [72b]
    Faith, C.: Modules finite over endomorphism ring. Proc. Tulane University, Symposium in Ring Theory, New Orleans. Lecture Notes in Mathematics, No. 246. Berlin/Heidelberg/New York: Springer, pp. 145–190.Google Scholar
  12. [62]
    Goldie, A. W.: Non-commutative principal ideal rings. Archiv. Math. 13, 213–221 (1962).CrossRefMathSciNetGoogle Scholar
  13. [58]
    Hochschild, G.: Note on relative homological algebra. Nagoya Math. J. 13, 89-94 (1958).zbMATHMathSciNetGoogle Scholar
  14. [58a]
    Kaplansky, I.: Projective modules. Ann. Math. 68, 372–377 (1958).CrossRefzbMATHMathSciNetGoogle Scholar
  15. [59]
    Kaplansky, I.: Homological dimension of rings and modules, Mimeographed notes. University of Chicago 1969.Google Scholar
  16. [66]
    Kaplansky, I.: The homological dimension of a quotient field. Nagoya Math. J. 27, 139–142 (1966).zbMATHMathSciNetGoogle Scholar
  17. [69a]
    Kaplansky, I.: Fields and Rings, Chicago Lectures in Mathematics. Chicago/ London: University of Chicago Press 1969.Google Scholar
  18. [69b]
    Kaplansky, I.: Infinite Abelian Groups, 2nd Ed. Ann Arbor: University of Michigan Press 1969.zbMATHGoogle Scholar
  19. [63a]
    Levy, L.: Unique direct sums of prime rings. Trans. Amer. Math. Soc. 106, 64–76 (1963).CrossRefzbMATHGoogle Scholar
  20. [63b]
    Levy, L.: Torsionfree and divisible modules over non-integral domains. Can. J. Math. 15, 132–151 (1963).zbMATHGoogle Scholar
  21. [63]
    MacLane, S.: Homology. Berlin/Göttingen/Heidelberg: Springer 1963.zbMATHGoogle Scholar
  22. [65]
    Mitchell, B.: Theory of Categories. New York: Academic Press 1965.zbMATHGoogle Scholar
  23. [60]
    Northcott, D. G.: Homological Algebra. Cambridge: Cambridge University Press 1960.zbMATHGoogle Scholar
  24. [68c]
    Osofsky, B. L.: Homological dimensions and the continuum hypothesis. Trans. Amer. Math. Soc. 132, 217–230 (1968).CrossRefzbMATHMathSciNetGoogle Scholar
  25. [69]
    Osofsky, B. L.: A commutative local ring with finite divisors. Trans. Amer. Math. Soc. 141, 377-385 (1969).CrossRefzbMATHMathSciNetGoogle Scholar
  26. [70]
    Osofsky, B. L.: Homological dimension and cardinality. Trans. Amer. Math. Soc. 151, 641–649 (1970).CrossRefzbMATHMathSciNetGoogle Scholar
  27. [65]
    Mitchell, B.: Theory of Categories. New York: Academic Press 1965.zbMATHGoogle Scholar
  28. [60]
    Northcott, D. G.: Homological Algebra. Cambridge: Cambridge University Press 1960.zbMATHGoogle Scholar
  29. [68c]
    Osofsky, B. L.: Homological dimensions and the continuum hypothesis. Trans. Amer. Math. Soc. 132, 217–230 (1968).CrossRefzbMATHMathSciNetGoogle Scholar
  30. [69]
    Osofsky, B. L.: A commutative local ring with finite divisors. Trans. Amer. Math. Soc. 141, 377-385 (1969).CrossRefzbMATHMathSciNetGoogle Scholar
  31. [70]
    Osofsky, B. L.: Homological dimension and cardinality. Trans. Amer. Math. Soc. 151, 641–649 (1970).CrossRefzbMATHMathSciNetGoogle Scholar
  32. [64]
    Sandomierski, F.: Relative injectivity and projectivity. Ph. D. Thesis, Penna, State University, University Park 1964.Google Scholar
  33. [70]
    Webber, D. B.: Ideals and modules of simple Noetherian hereditary rings. J. Algebra 16, 239–242 (1970).CrossRefzbMATHMathSciNetGoogle Scholar
  34. [08]
    Wedderburn, J. H. M.: On hypercomplex numbers. Proc. Lond. Math. Soc. (2) 6, 77–117 (1908).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag, Berlin · Heidelberg 1973

Authors and Affiliations

  • Carl Faith
    • 1
  1. 1.Rutgers UniversityNew BrunswickUSA

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