Algebras, Rings and Modules Non-commutative Algebras and Rings

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.

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

Michiel Hazewinkel, Nadiya M. Gubareni