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Algebra pp 110-185 | Cite as

Ring and Module

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

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.

Keywords

Exact Sequence Commutative Ring Left Ideal Injective Module Endomorphism 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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Copyright information

© Springer-Verlag, Berlin · Heidelberg 1973

Authors and Affiliations

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

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