Abstract
A list of topics in this chapter includes ring, module, endomorphism ring, biendomorphism ring (= bicentralizer) of a module, left ideals, nilpotency, simplicity in categories (e.g. simple groups, simple modules, simple rings), representations of simple rings in complete rings of linear transformations (the prelude to the Chevalley-Jacobson density theorem proved in Chapter 19), matrix rings, dual modules, dual basis lemma for projective modules, the “dual” theorem for generators of the category mod-R of all right R-modules, the trace of a module, additive categories and functors, exact sequences, left (right, half) exact functors, the left exactitude of the morphism (horn) functors h A: mod-R ➦ mod-ℤ and h A : mod-R ➦ mod-ℤ, for any object A of mod-R, Baer’s criterion for injectivity, enough injectives, and the existence of injective cogenerators of mod-R, fully faithful functors, idealizers, orthogonal idempotents and direct decompositions of modules, and fully invariant submodules of a module, including the radical and the socle.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
General References
Artin, E.: The influence of J. H. M. Wedderburn on the development of modern algebra. Bull. Amer. Math. Soc. 56, 65–72 (1950).
Artin, E., Nesbitt, E., Thrall, R.: Rings with Minimum Condition. Ann Arbor: University of Michigan Press 1944.
Baer, R.: Abelian groups that are direct summands of every containing Abelian group. Bull. Amer. Math. Soc. 46, 800–806 (1940).
Bass, H.: Algebraic K-Theory. New York: Benjamin 1968.
Bourbaki, N.: Éléments de Mathématiques, Livre II, Algèbre. Actualités Scienti-fiques et Industrielles, No. 1261. Paris: Hermann 1958, Chapter 8.
Brauer, R., Weiss, E.: Non-commutative Rings, Part I (Lecture Notes). Mathematics Department, Harvard University, Cambridge circa 1964.
Curtis, C., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. New York: Interscience Publishers, Wiley 1962.
Eckmann, B., Schopf, A.: Über injektive Moduln. Arch. Math. 4, 75–78 (1953).
Faith, C.: Lectures on Injective Modules and Quotient Rings. Berlin/Heidelberg/ New York: Springer 1967.
Herstein, I. N.: Noncommutative Rings. Carus Math. Monographs, No. 15. Amer. Math. Ass. and Wiley, New York 1968.
Jacobson, N.: The Theory of Rings, Surveys, Vol. 2. Amer. Math. Soc., Providence, 1942.
Jacobson, N.: Structure of Rings, Colloquium Publication, Vol. 37. Amer. Math. Soc., Providence 1956 (1964 revised).
Jans, J.: Rings and Homology. New York: Holt, Rinehart & Winston 1964.
Kaplansky, I.: Fields and Rings. Chicago: University of Chicago Press 1969.
Lambek, J.: Rings and Modules. New York: Blaisdell 1966.
MacLane, S.: Homology. Berlin/Göttingen/Heidelberg: Springer 1963.
MacLane, S.: Categories for the Working Mathematician, Springer 1971.
Mitchell, B.: Theory of Categories. New York: Academic Press 1965.
Morita, K.: Duality for modules and its application to the theory of rings with minimum condition. Sci. Reports, Tokyo Kyoiku Daigaku 6, 83–142 (1958).
Pontryagin, L.: Topological Groups. Princeton: Princeton University Press 1939.
Waerden, B. L. van der: Modern Algebra. New York: Ungar 1948.
Zariski, O., Samuel, P.: Commutative Algebra, Vol. I. Princeton/New York: Van Nostrand 1958.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1973 Springer-Verlag, Berlin · Heidelberg
About this chapter
Cite this chapter
Faith, C. (1973). Ring and Module. In: Algebra. Die Grundlehren der mathematischen Wissenschaften, vol 190. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80634-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-80634-6_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-80636-0
Online ISBN: 978-3-642-80634-6
eBook Packages: Springer Book Archive