Abstract
The study of endomorphism rings is important both in the study of modules and in the study of abstract rings. It is important in the study of modules because many properties of a module are characterizable by properties of its endomorphism ring. Indeed, some modules over certain rings are determined by their endomorphism rings, which is a way of saying that the modules are isomorphic if and only if their endomorphism rings are isomorphic qua rings. This is true for vector spaces. It is important for the study of rings, since some rings, even those abstractly defined, can be embedded in a ring of endomorphisms of a relatively “nice” module, for example, a vector space.
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
Gorenstein, D.: Finite Groups. New York: Harper & Row 1968.
Artin, E., Whaples, G.: The theory of simple rings. Amer. J. Math. 65, 87–107 (1943).
Artin, E., Nesbitt, E., Thrall, R.: Rings with Minimum Condition. Ann Arbor: University of Michigan Press 1944.
Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc. 95, 466–488 (1960).
Cohn, P. M.: On subsemigroups of free semigroups. Proc. Amer. Math. Soc. 13, 347–351 (1962).
Gorenstein, D.: Finite Groups. New York: Harper & Row 1968.
Lambek, J.: Torsion Theories, Additive Semantics, and Rings of Quotients. Lecture Notes in Mathematics, No. 177. Berlin/Heidelberg/New York: Springer 1971. Lambek, J., Findlay, G. D. (see Findlay and Lambek).
Osofsky, B. L.: Homological dimension and cardinality. Trans. Amer. Math. Soc. 151, 641–649 (1970).
Rao, M. L. R.: Azumaya, semisimple, and ideal algebras. Bull. Amer. Math. Soc. 78, 588–592 (1972). Reiner, I., Curtis, C. W. (see Curtis and Reiner).
Cohn, P. M.: On subsemigroups of free semigroups. Proc. Amer. Math. Soc. 13, 347–351 (1962).
Cohn, P. M.: Morita Equivalence and Duality. University of London, Bookstore, Queen Mary College, Mile End Road, London 1966.
Cozzens, J. H.: Homological properties of the ring of differential polynomials. Bull. Amer. Math. Soc. 76, 75-79 (1970).
Cozzens, J. H.: Simple principal left ideal domains. J. Alg. 23, 66–75 (1972).
Faith, C.: Noetherian simple rings. Bull. Amer. Math. Soc. 70, 730— 731 (1964).
Faith, C.: Lectures on Injective Modules and Quotient Rings. Lecture Notes in Mathematics, No. 49. Berlin/Heidelberg/New York: Springer 1967.
b] Faith, C.: A general Wedderburn theorem. Bull. Amer. Math. Soc. 73, 65–67 (1967).
a] Faith, C.: A correspondence theorem for projective modules and the structure of simple noetherian rings. Bull. Amer. Math. Soc. 77, 338–342 (1971).
Faith, C.: Orders in semilocal rings. Bull. Amer. Math. Soc. 77, 960–962 (1971).
Faith, C., Utumi, Y.: Maximal quotient rings. Proc. Amer. Math. Soc. 16, 1084-1089 (1965).
Findlay, G. D., Lambek, J.: A generalized ring of quotients, I, II. Canad. Math. Bull. 1, 77-85, 155–167 (1958).
Fuller, K. R.: Primary rings and double centralizers. Pac. J. Math. 34, 379-383 (1970). Fuller, K. R., Camillo, V. (see Camillo and Fuller).Fuller, K. R., Dickson, S. E. (see Dickson and Fuller).
Gabriel, P.: Des catégories abeliennes. Bull. Soc. Math. France 90, 323–448 (1962).
Jacobson, N.: The theory of rings. Surveys, Vol. 2. Amer. Math. Soc., Providence 1942.
Levitzki, J.: Solution of a problem of G. Koethe. Amer. J. Math. 67, 437–442 (1945).
Osofsky, B. L.: Homological dimension and cardinality. Trans. Amer. Math. Soc. 151, 641–649 (1970).
Seshadri, C.: Trivality of vector bundles over the affine space K2. Proc. Nat. Acad. Sci. USA 44 456–458 (1958).
Bergman, G. M.: A ring primitive on the right but not the left. Proc. Amer. Math. Soc. 15, 473–475 (1964).
Birkhoff, G.: Lattice Theory, Colloquium Publication, Vol. 25 (revised). Amer. Math. Soc., Providence 1967.
Cartan, H., Eilenberg, S.: Homological Algebra. Princeton: Princeton University Press 1956.
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). General Wedderburn Theorems. In: Algebra. Die Grundlehren der mathematischen Wissenschaften, vol 190. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80634-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-80634-6_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-80636-0
Online ISBN: 978-3-642-80634-6
eBook Packages: Springer Book Archive