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 | i ∈ I}. 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.)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Albrecht, F.: On projective modules over semi-hereditary rings. Proc. Amer. Math. Soc. 12, 638–639 (1961).
Artin, E.: Zur Theorie der hyperkomplexen Zahlen. Abh. Math. Sem. Univ. Hamburg 5, 251–260 (1927).
Artin, E.: Zur Arithmetik hyperkomplexer Zahlen. Abh. Math. Sem. Univ. Hamburg 5, 261–289 (1927).
Auslander, M.: On the dimension of modules and algebras, III. Nagoya Math. J. 9, 67–77 (1955).
Auslander, M., Buchsbaum, D.: Homological dimension in local rings. Trans. Amer. Math. Soc. 85, 390–405 (1957).
Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc. 95, 466–488 (1960).
Cartan, H., Eilenberg, S.: Homological Algebra. Princeton: Princeton University Press 1956.
Eisenbud, D., Griffith, P.: Serial rings. J. Algebra 17, 389–400 (1971).
Faith, C.: Rings with ascending condition on annihilators. Nagoya Math. J.27, 179–191 (1966).
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).
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.
Goldie, A. W.: Non-commutative principal ideal rings. Archiv. Math. 13, 213–221 (1962).
Hochschild, G.: Note on relative homological algebra. Nagoya Math. J. 13, 89-94 (1958).
Kaplansky, I.: Projective modules. Ann. Math. 68, 372–377 (1958).
Kaplansky, I.: Homological dimension of rings and modules, Mimeographed notes. University of Chicago 1969.
Kaplansky, I.: The homological dimension of a quotient field. Nagoya Math. J. 27, 139–142 (1966).
Kaplansky, I.: Fields and Rings, Chicago Lectures in Mathematics. Chicago/ London: University of Chicago Press 1969.
Kaplansky, I.: Infinite Abelian Groups, 2nd Ed. Ann Arbor: University of Michigan Press 1969.
Levy, L.: Unique direct sums of prime rings. Trans. Amer. Math. Soc. 106, 64–76 (1963).
Levy, L.: Torsionfree and divisible modules over non-integral domains. Can. J. Math. 15, 132–151 (1963).
MacLane, S.: Homology. Berlin/Göttingen/Heidelberg: Springer 1963.
Mitchell, B.: Theory of Categories. New York: Academic Press 1965.
Northcott, D. G.: Homological Algebra. Cambridge: Cambridge University Press 1960.
Osofsky, B. L.: Homological dimensions and the continuum hypothesis. Trans. Amer. Math. Soc. 132, 217–230 (1968).
Osofsky, B. L.: A commutative local ring with finite divisors. Trans. Amer. Math. Soc. 141, 377-385 (1969).
Osofsky, B. L.: Homological dimension and cardinality. Trans. Amer. Math. Soc. 151, 641–649 (1970).
Mitchell, B.: Theory of Categories. New York: Academic Press 1965.
Northcott, D. G.: Homological Algebra. Cambridge: Cambridge University Press 1960.
Osofsky, B. L.: Homological dimensions and the continuum hypothesis. Trans. Amer. Math. Soc. 132, 217–230 (1968).
Osofsky, B. L.: A commutative local ring with finite divisors. Trans. Amer. Math. Soc. 141, 377-385 (1969).
Osofsky, B. L.: Homological dimension and cardinality. Trans. Amer. Math. Soc. 151, 641–649 (1970).
Sandomierski, F.: Relative injectivity and projectivity. Ph. D. Thesis, Penna, State University, University Park 1964.
Webber, D. B.: Ideals and modules of simple Noetherian hereditary rings. J. Algebra 16, 239–242 (1970).
Wedderburn, J. H. M.: On hypercomplex numbers. Proc. Lond. Math. Soc. (2) 6, 77–117 (1908).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1973 Springer-Verlag, Berlin · Heidelberg
About this chapter
Cite this chapter
Faith, C. (1973). Semisimple Modules and Homological Dimension. In: Algebra. Die Grundlehren der mathematischen Wissenschaften, vol 190. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80634-6_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-80634-6_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-80636-0
Online ISBN: 978-3-642-80634-6
eBook Packages: Springer Book Archive