Algebra pp 110-185 | Cite as

Ring and Module

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


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.


Exact Sequence Commutative Ring Left Ideal Injective Module Endomorphism Ring 
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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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