Algebras, Rings and Modules : Non-commutative Algebras and Rings book cover
1st Edition

Algebras, Rings and Modules
Non-commutative Algebras and Rings

ISBN 9781482245035
Published January 26, 2016 by CRC Press
388 Pages 10 B/W Illustrations

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Book Description

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.

Table of Contents



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


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



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Rings, which play a fundamental role in analysis, geometry, and topology, constitute perhaps the most ubiquitous algebraic structures across mathematics. Starting with fields (the most familiar rings), considering polynomials leads to commutative rings and considering matrices leads to non-commutative algebra. The present volume, though lacking a number, joins a series now spread over three publishers. Like volumes 1 and 2, it surveys aspects of non-commutative rings (volume 3 went in another direction) but stands self-contained, despite the occasional reference to previous volumes. Progress in ring theory depends on conditions designed to isolate special classes of rings admitting satisfying structure theorems. Each chapter surveys such conditions and their consequences: hereditary rings, valuation domains, nonsingular rings, Goldie rings, FDI-rings, exchange rings, Rickart rings, serial nonsingular rings, and many more. Commutative ring concepts tend to have diverse, but limited, generalizations in the non-commutative context, and the early chapters particularly explore such themes. The authors supply complete details, even for simple arguments that other authors might package into exercises (of which this book has none), making the book an excellent reference. Readers will find all developments digested into small satisfying steps, with major results seeming to drop out effortlessly.

--D. V. Feldman, University of New Hampshire, Appeared in February 2017 issue of CHOICE