Ring theory. Volume I.

*(English)*Zbl 0651.16001
Pure and Applied Mathematics, 127. Boston, MA etc.: Academic Press, Inc. xxiv, 538 p. $ 89.50 (1988).

From the foreword: “The principal object of these two volumes is to present the mainstream of ring theory, being a source both of results and proofs. The first volume deals with general structure results and could serve as an introductory text; the second focuses on the classes of rings that have occupied most attention in the literature.”

The book under review presents an excellent addition to such well known ring books as “The Theory of Rings” and “Structure of Rings” by Jacobson, “Noncommutative Rings” by Herstein, and “Algebra: Rings, Modules and Categories I”, “Algebra II, Ring Theory” by Faith. Though containing a good portion of what may be called the traditional ring theory (especially in Volume I), Rowen’s book deals with many ring topics not considered in the monographs mentioned. Some of these topics are: the representation theory of finite dimensional algebras over a field, including the Brauer-Thrall conjecture, almost split sequences, Gabriel quiver and The Multiplicative Basis theorem; dimension theory (Krull dimensions in the noncommutative case, Gabriel dimension, Goldie dimension, homological dimensions, Gelfand-Kirillov dimension); Serre’s conjecture; the Merkur’ev-Suslin theorem on central simple algebras; group algebras and enveloping algebras of Lie algebras (the last two topics are considered both as “Rings from representation theory”). There are also many ring notions which are treated in the present book in much more completeness than in the books by Jacobson, Herstein, Faith (e.g. skew polynomial rings, quotient rings, Noetherian rings, affine algebras). Some ring theories included in Faith’s books in exercises are discussed in the book under review in details (e.g. Jategaonkar’s examples of one-side-primitive rings). The whole Chapter 5 deals with homology and cohomology (diagrams, resolutions, homological dimensions etc.)

The author uses the following hierarchy: Main Text, Supplementary material, Appendices, Exercises, Digressions. The meaning of such division may be illustrated as follows. In Chapter 3 “Rings of Fractions and Embedding Theorems” the Goldie theorems are treated in Main Text, Embeddings in special types of rings are considered in Supplements, orders in semilocal rings are discussed in Digression. The book contains also many exercises in which some interesting and important theorems are formulated. In exercises are also given brief sketches of some ring theories, e.g. Quasi-Frobenius rings.

It should be stressed that by a ring the author means a ring with identity element (as Faith does); accordingly homomorphisms of a ring are defined. A ring without identity the author calls “rng”, and the appropriate category is denoted by “Rng”.

The reviewer has noticed some typos, e.g. Volume I, page 23, the 7-th line from above, “Corollary 0.2.21” is printed instead of “Corollary 0.2.20”, and the text is wrong. The definition of irreducible ideal (Volume I, page 375, the last two lines) is unclear. There are other typos but their correction is obvious.

The detailed Table of Contents given below shows topics considered in Volume I. We add to it some remarks.

Chapter 0 deals with elementary basic notions: rings, modules, categories etc.

In Chapter 1 the most important ring constructions are considered (matrix rings, polynomial rings, products, sums etc.), each construction being given with a small theory in which the corresponding properties are discussed. The chapter contains many generalizations of polynomial rings (Ore extensions, skew polynomial rings and Laurent series together with their “skew” versions, differential polynomial rings including the Weil algebra).

Chapter 2 is devoted to the basic structure theory. Here the Jacobson radical and the nilradicals are considered, and the theory of such important classes of rings as prime and semiprime rings, primitive rings, perfect and semiperfect rings is discussed. Some basic module notions (projectivity, injectivity, indecomposability) are treated. A special consideration is given to the Brauer-Thrall conjecture on algebras over a field of finite representation type, and the proof of Rojter’s theorem (the positive solution of the first part of the Brauer-Thrall conjecture) is given.

Chapter 3 deals with rings of fractions and embedding theorems. The author says: “The main theme of this chapter is embedding a ring into a nicer ring which can be studied more easily.” As such “nice rings” matrix rings, division rings, and Artinian rings are taken. Some types of quotient rings are considered (rings of fractions, Johnson’s ring of quotients, Martindale-Amitsur ring of quotients, the maximal ring of quotients). Localization (including the noncommutative case) is considered too.

More than one third of the chapter is devoted to left Noetherian rings, including heights of prime ideals, Jacobson’s conjecture on the intersection of powers of the Jacobson radical of a Noetherian ring, and the Krull dimension in the noncommutative case.

Chapter 4 deals mostly with Morita theorems.

Volume I contains also the Magnus-Witt theorem on factors of the lower central series of a free group (Theorem 1.3.38; the proof is given in Appendix A). Appendix B contains some basic elementary information on Banach algebras.

Contents (Volume I): Foreword; Introduction: an Overview of Ring Theory; Table of Principal Notation; Chapter 0, General Fundamentals: 0.0. Preliminary Foundations, 0.1 Categories of Rings and Modules, 0.2 Finitely Generated Modules, Simple Modules and Artin Modules, 0.3 Abstract Dependence, Exercises; Chapter 1, Construction of Rings: 1.1 Matrix Rings and Idempotents, 1.2 Polynomial Rings, 1.3. Free Modules and Rings, 1.4 Products and Sums, 1.5 Endomorphism Rings and Regular Representation, 1.6 Automorphisms, Derivations, and Skew Polynomial Rings, 1.7 Tensor Products, 1.8 Direct Limits and Inverse Limits, 1.9 Graded Rings and Modules, 1.10 Central Localization, Exercises; Chapter 2, Basic Structure Theory: 2.1 Primitive Rings, 2.2 The Chinese Remainder Theorem and Subdirect Products, 2.3. Modules with Composition Series and Artinian Rings, 2.4 Completely Reducible Modules and the Socle, 2.5 The Jacobson Radical, 2.6 Nilradicals, 2.7 Semiprimary Rings and Their Generalizations, 2.8 Projective Modules (An Introduction), 2.9 Indecomposable Modules and LE-Modules, 2.10 Injective Modules, 2.11 Exact Functors, 2.12 The Prime Spectrum, 2.13 Rings with Involution, Exercises; Chapter 3, Rings of Fractions and Embedding Theorems: 3.1 Classical Rings of Fractions, 3.2 Goldie’s Theorems and Orders in Artinian Quotient Rings, 3.3 Localization of Nonsingular Rings and Their Modules, 3.4 Noncommutative Localization, 3.5 Left Noetherian Rings, Exercises; Chapter 4, Categorical Aspects of Module Theory: 4.1 The Morita Theorems, 4.2 Adjoints, Exercises; Appendix A: The Proof of Magnus-Witt Theorem, Exercises; Appendix B: Normed Algebras and Banach Algebras, Exercises; The Basic Ring-Theoretic Notions and Their Characterizations; Major Ring- and Module-Theoretic Results Proved in Volume I; References; Subject Index.

The book under review presents an excellent addition to such well known ring books as “The Theory of Rings” and “Structure of Rings” by Jacobson, “Noncommutative Rings” by Herstein, and “Algebra: Rings, Modules and Categories I”, “Algebra II, Ring Theory” by Faith. Though containing a good portion of what may be called the traditional ring theory (especially in Volume I), Rowen’s book deals with many ring topics not considered in the monographs mentioned. Some of these topics are: the representation theory of finite dimensional algebras over a field, including the Brauer-Thrall conjecture, almost split sequences, Gabriel quiver and The Multiplicative Basis theorem; dimension theory (Krull dimensions in the noncommutative case, Gabriel dimension, Goldie dimension, homological dimensions, Gelfand-Kirillov dimension); Serre’s conjecture; the Merkur’ev-Suslin theorem on central simple algebras; group algebras and enveloping algebras of Lie algebras (the last two topics are considered both as “Rings from representation theory”). There are also many ring notions which are treated in the present book in much more completeness than in the books by Jacobson, Herstein, Faith (e.g. skew polynomial rings, quotient rings, Noetherian rings, affine algebras). Some ring theories included in Faith’s books in exercises are discussed in the book under review in details (e.g. Jategaonkar’s examples of one-side-primitive rings). The whole Chapter 5 deals with homology and cohomology (diagrams, resolutions, homological dimensions etc.)

The author uses the following hierarchy: Main Text, Supplementary material, Appendices, Exercises, Digressions. The meaning of such division may be illustrated as follows. In Chapter 3 “Rings of Fractions and Embedding Theorems” the Goldie theorems are treated in Main Text, Embeddings in special types of rings are considered in Supplements, orders in semilocal rings are discussed in Digression. The book contains also many exercises in which some interesting and important theorems are formulated. In exercises are also given brief sketches of some ring theories, e.g. Quasi-Frobenius rings.

It should be stressed that by a ring the author means a ring with identity element (as Faith does); accordingly homomorphisms of a ring are defined. A ring without identity the author calls “rng”, and the appropriate category is denoted by “Rng”.

The reviewer has noticed some typos, e.g. Volume I, page 23, the 7-th line from above, “Corollary 0.2.21” is printed instead of “Corollary 0.2.20”, and the text is wrong. The definition of irreducible ideal (Volume I, page 375, the last two lines) is unclear. There are other typos but their correction is obvious.

The detailed Table of Contents given below shows topics considered in Volume I. We add to it some remarks.

Chapter 0 deals with elementary basic notions: rings, modules, categories etc.

In Chapter 1 the most important ring constructions are considered (matrix rings, polynomial rings, products, sums etc.), each construction being given with a small theory in which the corresponding properties are discussed. The chapter contains many generalizations of polynomial rings (Ore extensions, skew polynomial rings and Laurent series together with their “skew” versions, differential polynomial rings including the Weil algebra).

Chapter 2 is devoted to the basic structure theory. Here the Jacobson radical and the nilradicals are considered, and the theory of such important classes of rings as prime and semiprime rings, primitive rings, perfect and semiperfect rings is discussed. Some basic module notions (projectivity, injectivity, indecomposability) are treated. A special consideration is given to the Brauer-Thrall conjecture on algebras over a field of finite representation type, and the proof of Rojter’s theorem (the positive solution of the first part of the Brauer-Thrall conjecture) is given.

Chapter 3 deals with rings of fractions and embedding theorems. The author says: “The main theme of this chapter is embedding a ring into a nicer ring which can be studied more easily.” As such “nice rings” matrix rings, division rings, and Artinian rings are taken. Some types of quotient rings are considered (rings of fractions, Johnson’s ring of quotients, Martindale-Amitsur ring of quotients, the maximal ring of quotients). Localization (including the noncommutative case) is considered too.

More than one third of the chapter is devoted to left Noetherian rings, including heights of prime ideals, Jacobson’s conjecture on the intersection of powers of the Jacobson radical of a Noetherian ring, and the Krull dimension in the noncommutative case.

Chapter 4 deals mostly with Morita theorems.

Volume I contains also the Magnus-Witt theorem on factors of the lower central series of a free group (Theorem 1.3.38; the proof is given in Appendix A). Appendix B contains some basic elementary information on Banach algebras.

Contents (Volume I): Foreword; Introduction: an Overview of Ring Theory; Table of Principal Notation; Chapter 0, General Fundamentals: 0.0. Preliminary Foundations, 0.1 Categories of Rings and Modules, 0.2 Finitely Generated Modules, Simple Modules and Artin Modules, 0.3 Abstract Dependence, Exercises; Chapter 1, Construction of Rings: 1.1 Matrix Rings and Idempotents, 1.2 Polynomial Rings, 1.3. Free Modules and Rings, 1.4 Products and Sums, 1.5 Endomorphism Rings and Regular Representation, 1.6 Automorphisms, Derivations, and Skew Polynomial Rings, 1.7 Tensor Products, 1.8 Direct Limits and Inverse Limits, 1.9 Graded Rings and Modules, 1.10 Central Localization, Exercises; Chapter 2, Basic Structure Theory: 2.1 Primitive Rings, 2.2 The Chinese Remainder Theorem and Subdirect Products, 2.3. Modules with Composition Series and Artinian Rings, 2.4 Completely Reducible Modules and the Socle, 2.5 The Jacobson Radical, 2.6 Nilradicals, 2.7 Semiprimary Rings and Their Generalizations, 2.8 Projective Modules (An Introduction), 2.9 Indecomposable Modules and LE-Modules, 2.10 Injective Modules, 2.11 Exact Functors, 2.12 The Prime Spectrum, 2.13 Rings with Involution, Exercises; Chapter 3, Rings of Fractions and Embedding Theorems: 3.1 Classical Rings of Fractions, 3.2 Goldie’s Theorems and Orders in Artinian Quotient Rings, 3.3 Localization of Nonsingular Rings and Their Modules, 3.4 Noncommutative Localization, 3.5 Left Noetherian Rings, Exercises; Chapter 4, Categorical Aspects of Module Theory: 4.1 The Morita Theorems, 4.2 Adjoints, Exercises; Appendix A: The Proof of Magnus-Witt Theorem, Exercises; Appendix B: Normed Algebras and Banach Algebras, Exercises; The Basic Ring-Theoretic Notions and Their Characterizations; Major Ring- and Module-Theoretic Results Proved in Volume I; References; Subject Index.

Reviewer: J.S.Ponizovskij

##### MSC:

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16-XX | Associative rings and algebras |

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |

16P50 | Localization and associative Noetherian rings |

16N60 | Prime and semiprime associative rings |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |

16Nxx | Radicals and radical properties of associative rings |

16P40 | Noetherian rings and modules (associative rings and algebras) |

16P20 | Artinian rings and modules (associative rings and algebras) |

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

16E10 | Homological dimension in associative algebras |

16Gxx | Representation theory of associative rings and algebras |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

16D50 | Injective modules, self-injective associative rings |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |