This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.
Preface
Symbols
Elementary properties of rings
Basic notions
Special elements and ideals in rings
Special rings
Chain conditions for rings
Algebras and group rings
Module categories
Elementary properties of modules
The category of R-modules
Internal direct sum
Product, coproduct and subdirect product
Pullback and pushout
Functors, Hom-functors
Tensor product, tensor functor
Modules characterized by the Hom-functor
Generators, trace
Congenerators, reject
Subgenerators, the category o [M]
Injective modules
Essential extensions, injective hulls
Projective modules
Superfluous epimorphisms, projective covers
Notions derived from simple modules
Semisimple modules and rings
Socle and radical of modules and rings
The radical of endomorphism rings
Co-semisimple and good modules and rings
Finiteness conditions in modules
The direct limit
Finitely presented modules
Coherent modules and rings
Noetherian modules and rings
Annihilator conditions
Dual finiteness conditions
The inverse limit
Finitely copresented modules
Artinian and co-noetherian modules
Modules of finite length
Pure sequences and derived notions
P-pure sequences, pure projective modules
Purity in o[M], R-MOD and ZZ-MOD
Absolutely pure modules
Flat modules
Regular modules and rings
Copure sequences and derived notions
Modules described by means of projectivity
(Semi)hereditary modules and rings
Semihereditary and hereditary domains
Supplemented modules
Semiperfect modules and rings
Perfect modules and rings
Relations between functors
Functional morphisms
Adjoint pairs of functors
Equivalences of categories
Dualities between categories
Quasi-Frobenius modules and rings
Functor rings
Rings with local units
Global dimensions of modules and rings
The functor Hom(V,-)
Functor rings of o[M] and R-MOD
Pure semisimple modules and rings
Modules of finite representation type
Serial modules and rings
Homo-serial modules and rings
Bibliography
Index
Biography
Wisbauer, Robert
