E-Book, Englisch, 388 Seiten
Hazewinkel / Gubareni Algebras, Rings and Modules
1. Auflage 2015
ISBN: 978-1-4822-4505-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
Non-commutative Algebras and Rings
E-Book, Englisch, 388 Seiten
ISBN: 978-1-4822-4505-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
The theory of algebras, rings, and modules is one of the fundamental domains of modern mathematics. General algebra, more specifically non-commutative algebra, is poised for major advances in the twenty-first century (together with and in interaction with combinatorics), just as topology, analysis, and probability experienced in the twentieth century. This volume is a continuation and an in-depth study, stressing the non-commutative nature of the first two volumes of Algebras, Rings and Modules by M. Hazewinkel, N. Gubareni, and V. V. Kirichenko. It is largely independent of the other volumes. The relevant constructions and results from earlier volumes have been presented in this volume.
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
Preface
Preliminaries
Basic concepts of rings and modules
Categories and functors
Tensor product of modules
Direct and inverse limits
Projective, injective and °at modules
The functor Tor
The functor Ext
Semiperfect and perfect rings
Serial and semidistributive rings
Classical rings of fractions
Quivers of rings
Basic general constructions of rings and modules
Direct and semidirect products
Group rings, smash and crossed products
Polynomial and skew polynomial rings
Formal power and skew power series rings
Laurent polynomial and series rings
Generalized matrix rings. Generalized triangular matrix rings
G-graded rings
Notes and references
Valuation rings
Valuation domains
Discrete valuation domains
Valuation rings of division rings
Discrete valuation rings of division rings
Other types of valuation rings
Approximation theorem for valuation rings
Notes and references
Homological dimensions of rings and modules
Projective and injective dimensions
Flat and weak dimensions
Homological characterization of some classes of rings
Torsionless modules
Flat modules and coherent rings
Modules over formal triangular matrix rings
Notes and references
Goldie and Krull dimensions of rings and modules
Uniform modules and uniform dimension
Injective uniform modules
Nonsingular modules and rings
Nonsingular rings and Goldie rings
Reduced rank and Artinian quotient rings
Krull dimension
Notes and references
Rings with Finiteness conditions
Some examples of Noetherian rings
Dedekind-finite rings and stable finite rings
FDI-rings
Semiprime FDI-rings
Notes and references
Krull-Remak-Schmidt-Azumaya theorem
The exchange property
The Azumaya theorem
Cancelation property
Exchange rings
Notes and references
Hereditary and semihereditary rings
Piecewise domains
Rickart rings and Small's theorems
Dimensions of hereditary and semihereditary rings
Right hereditary prime rings
Piecewise domains. Right hereditary perfect rings
Primely triangular matrix rings. The structure of piecewise domains
Right hereditary triangular rings
Noetherian hereditary primely triangular rings
Right hereditary species and tensor algebras
Notes and references
Serial nonsingular rings. Jacobson's conjecture
Structure of serial right Noetherian piecewise domains
Structure of serial nonsingular rings
Serial rings with Noetherian diagonal
Krull intersection theorem
Jacobson's conjecture
Notes and references
Rings related to Finite posets
Incidence rings
Incidence rings I(S;D)
Right hereditary rings A(S;O)
Incidence rings modulo radical
Serial and semidistributive rings I(S;¤;M)
Notes and references
Distributive and semidistributive rings
Distributive modules and rings
Semiprime semidistributive rings
Semiperfect semidistributive rings
Right hereditary SPSD-rings
Semihereditary SPSD-rings
Notes and references
The group of extensions
Module constructions pushout and pullback
The snake lemma
Extensions of modules
Baer sum of extensions
Properties of Ext1
Ext1 and extensions
Additive and Abelian categories
Notes and references
Modules over semiperfect rings
Finitely generated modules over semiperfect rings
Stable equivalence
Auslander-Bridger duality
Almost split sequences
Natural identities for Finitely presented modules
Almost split sequences over semiperfect rings
Linkage and duality of modules over semiperfect rings
Notes and references
Representations of primitive posets
Representations of Finite posets
Main canonical forms of matrix problems
Trichotomy lemma
The Kleiner lemma
The main construction
Primitive posets of the infinite representation type
Primitive posets of the Finite representation type
Notes and references
Representations of quivers, species and finite dimensional algebras
Finite quivers and their representations
Species and their representations
Finite dimensional algebras of the finite representation type
Notes and references
Artinian rings of finite representation type
Eisenbud-Gri±th's theorem
Auslander's theorem for right Artinian rings
Artinian semidistributive rings
Artinian hereditary semidistributive rings of finite representation type
Notes and references
Semiperfect rings of bounded representation type
Semiperfect rings of bounded representation type
Modules over right hereditary SPSD-rings
Reduction of f.p. modules to mixed matrix problems
Some mixed matrix problems
Right hereditary SPSD-rings of unbounded representation type
Right hereditary SPSD-rings of bounded representation type
(K;O)-species and tensor algebras
(K;O)-species of bounded representation type
Notes and references
Bibliography
Index
