**Mathematical Surveys and Monographs**

Volume: 65;
2004;
475 pp;
Softcover

MSC: Primary 00; 01; 12; 13; 16;
Secondary 03; 06; 08; 14; 15; 18

Print ISBN: 978-0-8218-3672-9

Product Code: SURV/65.R

List Price: $110.00

Individual Member Price: $88.00

**Electronic ISBN: 978-1-4704-1292-0
Product Code: SURV/65.R.E**

List Price: $110.00

Individual Member Price: $88.00

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# Rings and Things and a Fine Array of Twentieth Century Associative Algebra: Second Edition

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*Carl Faith*

This book surveys more than 125 years of aspects of associative algebras, especially ring and module theory. It is the first to probe so extensively such a wealth of historical development. Moreover, the author brings the reader up to date, in particular through his report on the subject in the second half of the twentieth century.

Included in the book are certain categorical properties from theorems of Frobenius and Stickelberger on the primary decomposition of finite Abelian groups; Hilbert's basis theorem and his Nullstellensatz, including the modern formulations of the latter by Krull, Goldman, and others; Maschke's theorem on the representation theory of finite groups over a field; and the fundamental theorems of Wedderburn on the structure of finite dimensional algebras and finite skew fields and their extensions by Braver, Kaplansky, Chevalley, Goldie, and others. A special feature of the book is the in-depth study of rings with chain condition on annihilator ideals pioneered by Noether, Artin, and Jacobson and refined and extended by many later mathematicians.

Two of the author's prior works,

In addition to the mathematical survey, the author gives candid and descriptive impressions of the last half of the twentieth century in “Part II: Snapshots of Some Mathematical Friends and Places”. Beginning with his teachers and fellow graduate students at the University of Kentucky and at Purdue, Faith discusses his Fulbright-NATO Postdoctoral at Heidelberg and at the Institute for Advanced Study (IAS) at Princeton, his year as a visiting scholar at Berkeley, and the many acquaintances he met there and in subsequent travels in India, Europe, and most recently, Barcelona.

“Researchers in algebra should find it both enjoyable to read and very useful in their work. In all cases, [Faith] cites full references as to the origin and development of the theorem .... I know of no other work in print which does this as thoroughly and as broadly.”

—John O'Neill, University of Detroit at Mercy

“ ‘Part II: Snapshots of Some Mathematical Friends and Places’ is wonderful! [It is] a joy to read! Mathematicians of my age and younger will relish reading ‘Snapshots’.”

—James A. Huckaba, University of Missouri-Columbia

#### Table of Contents

# Table of Contents

## Rings and Things and a Fine Array of Twentieth Century Associative Algebra: Second Edition

Table of Contents pages: 1 2 3 4

- Contents vii8 free
- Symbols xxiii24 free
- Preface to the Second Edition xxv26 free
- Acknowledgements to the Second Edition xxvii28 free
- Preface to the First Edition xxix30 free
- Acknowledgements to the First Edition xxxv36 free
- Part I. An Array of Twentieth Century Associative Algebra 138 free
- Chapter 1. Direct Product and Sums of Rings and Modules and the Structure of Fields 340
- §1.1 General Concepts 340
- §1.2 Internal Direct Sums 441
- §1.3 Products of Rings and Central Idempotents 542
- §1.4 Direct Summands and Independent Submodules 542
- §1.5 Dual Modules and Torsionless Modules 542
- §1.6 Torsion Abelian Groups 643
- §1.7 Primary Groups 643
- §1.8 Bounded Order 643
- §1.9 Theorems of Zippin and Frobenius-Stickelberger 643
- §1.10 Divisible Groups 744
- §1.11 Splitting Theorem for Divisible Groups 744
- §1.12 Second Splitting Theorem 744
- §1.13 Decomposition Theorem for Division Groups 744
- §1.14 Torsion Group Splits Off Theorem 744
- §1.15 Fundamental Theorem of Abelian Groups and Kulikoff's Subgroup Theorem 845
- §1.16 Corner's Theorem and the Dugas-Göbel Theorem 845
- §1.17 Direct Products as Summands of Direct Sums 845
- §1.18 Baer's Theorem 946
- §1.19 Specker-Nöbeling-Balcerzyk Theorems 946
- §1.20 Dubois' Theorem 946
- §1.21 Balcerzyk, Bialynicki, Birula and Los Theorem, Nunke's Theorem, and O'Neill's Theorem 946
- §1.22 Direct Sums as Summands of Their Direct Product 1047
- §1.23 Camillo's Theorem 1047
- §1.24 Lenzing's Theorem 1047
- §1.25 Zimmermann's Theorem on Pure Infective Modules 1047
- §1.26 Szele-Fuchs-Ayoub-Huynh Theorems 1047
- §1.27 Kertész-Huynh-Tominaga Torsion Splitting Theorems 1148
- §1.28 Three Theorems of Steinitz on the Structure of Fields 1148
- §1.29 Lüroth's Theorem 1350
- §1.30 Artin-Schreier Theory of Formally Real Fields 1350
- §1.31 Theorem of Castelnuovo-Zariski 1451
- §1.32 Monotone Minimal Generator Functions 1552
- §1.33 Quigley's Theorem: Maximal Subfields without α 1552

- Chapter 2. Introduction to Ring Theory: Schur's Lemma and Semisimple Rings, Prime and Primitive Rings, Noetherian and Artinian Modules, Nil, Prime and Jacobson Radicals 1754
- Quaternions 1754
- Hilbert's Division Algebra 1855
- When Everybody Splits 1855
- Artinian Rings and the Hopkins-Levitzki Theorem 1956
- Automorphisms of Simple Algebras: The Theorem of Skolem-Noether 2057
- Wedderburn Theory of Simple Algebras 2158
- Crossed Products and Factor Sets 2158
- Primitive Rings 2259
- Nil Ideals and the Jacobson Radical 2259
- The Chevalley-Jacobson Density Theorem 2259
- Semiprimitive Rings 2360
- Semiprimitive Polynomial Rings 2360
- Matrix Algebraic Algebras 2360
- Primitive Polynomial Rings 2461
- The Structure of Division Algebras 2562
- Tsen's Theorem 2562
- Cart an-Jacobson Galois Theory of Division Rings 2562
- Historical Note: Artin's Question 2663
- Jacobson a[sup(n(a))] = a Theorems and Kaplansky's Generalization 2663
- Kaplansky's Characterization of Radical Field Extensions 2764
- Radical Extensions of Rings 2764
- The Cartan-Brauer-Hua Theorem on Conjugates in Division Rings 2966
- Hua's Identity 2966
- Amitsur's Theorem and Conjugates in Simple Rings 3067
- Invariant Subrings of Matrix Rings 3168
- Rings Generated by Units 3168
- Transvections and Invariance 3269
- Other Commutativity Theorems 3269
- Noetherian and Artinian Modules 3370
- The Maximum and Minimum Conditions 3370
- Inductive Sets and Zorn's Lemma 3370
- Subdirectly Irreducible Modules: Birkhoff's Theorem 3471
- Jordan-Hölder Theorem for Composition Series 3572
- Two Noether Theorems 3572
- Hilbert Basis Theorem 3673
- Hilbert's Fourteenth Problem: Nagata's Solution 3774
- Noether's Problem in Galois Theory: Swan's Solution 3774
- Realizing Groups as Galois Groups 3774
- Prime Rings and Ideals 3875
- Chains of Prime Ideals 3976
- The Principal Ideal Theorems and the DCC on Prime Ideals 3976
- Primary and Radical Ideals 3976
- Lasker-Noether Decomposition Theorem 4077
- Hilbert Nullstellensatz 4178
- Prime Radical 4279
- Nil and Nilpotent Ideals 4380
- Nil Radicals 4481
- Simple Radical and Nil Rings 4582
- Semiprime Ideals and Unions of Prime Ideals 4582
- Maximal Annihilator Ideals Are Prime 4582
- Rings with Ace on Annihilator Ideals 4683
- The Baer Lower Nil Radical 4784
- Group Algebras over Formally Real Fields 4885
- Jacobson's Conjecture for Group Algebras 4986
- Simplicity of the Lie and Jordan Rings of Associative Rings: Herstein's Theorems 4986
- Simple Rings with Involution 4986
- Symmetric Elements Satisfying Polynomial Identities 5087
- Historical Notes 5188
- Separable Fields and Algebras 5188
- Wedderburn's Principal or Factor Theorem 5289
- Invariant Wedderburn Factors 5289

- Chapter 3. Direct Decompositions of Projective and Injective Modules 5390
- Direct Sums of Countably Generated Modules 5390
- Injective Modules and the Injective Hull 5491
- Injective Hulls: Baer's and Eckmann-Schopf's Theorems 5491
- Complement Submodules and Maximal Essential Extensions 5491
- The Cantor-Bernstein Theorem for Injectives 5592
- Generators and Cogenerators of Mod-R 5592
- Minimal Cogenerators 5693
- Cartan-Eilenberg, Bass, and Matlis-Papp Theorems 5693
- Two Theorems of Chase 5794
- Sets vs. Classes of Modules: The Faith-Walker Theorems 5794
- Polynomial Rings over Self-inject ive or QF Rings 5895
- Σ-injective Modules 5996
- Quasi-injective Modules and the Johnson-Wong Theorem 5996
- Dense Rings of Linear Transformations and Primitive Rings Revisited 6097
- Harada-Ishii Double Annihilate Theorem 6198
- Double Annihilator Conditions for Cogenerators 6198
- Koehler's and Boyle's Theorems 6299
- Quasi-injective Hulls 6299
- The Teply-Miller Theorem 63100
- Semilocal and Semiprimary Rings 63100
- Regular Elements and Ore Rings 63100
- Finite Goldie Dimension and Goldie's Theorem 64101
- The Wedderburn-Artin Theorem Revisited 64101
- The Faith-Utumi Theorem 65102
- Goldie's Principal Ideal Ring Theorem 65102
- Cailleau's Theorem 65102
- Local Rings and Chain Rings 66103
- Uniform Submodules and Maximal Complements 66103
- Beck's Theorems 67104
- Dade's Theorem 68105
- When Cyclic Modules Are Injective 68105
- When Simple Modules Are Inject ive: V-Rings 69106
- Cozzens' V-Domains 70107
- Projective Modules over Local or Semilocal Rings, or Semihereditary Rings 70107
- Serre's Conjecture, the Quillen-Suslin Solution and Seshadri's Theorem 71108
- Bass' Theorem on When Big Projectives Are Free 71108
- Projective Modules over Semiperfect Rings 72109
- Bass' Perfect Rings 72109
- Theorems of Björk and Jonah 73110
- Max Ring Theorems of Hamsher, Koifman, and Renault 73110
- Flat Covers Exist 74111
- The Socle Series of a Module and Loewy Modules 74111
- Semi-Artinian Rings and Modules 74111
- The Perlis Radical and the Jacobson Radical 75112
- The Frattini Subgroup of a Group 75112
- Krull's Intersection Theorem and Jacobson's Conjecture 75112
- Nakayama's Lemma 76113
- The Jacobson Radical and Jacobson-Hilbert Rings 76113
- Fully Bounded and FBN Rings 77114
- When Nil Implies Nilpotency 78115
- Shock's Theorem 78115
- Kurosch's Problem 79116
- The Nagata-Higman Theorem 79116
- N[sub(0)]-Categorical Nil Rings Are Nilpotent 79116
- The Golod-Shafarevitch Theorem 79116
- Some Amitsur Theorems on the Jacobson Radical 80117
- Köethe's Radical and Conjecture 80117
- A General Wedderburn Theorem 81118
- Koh's Schur Lemma 82119
- Categories 82119
- Morita's Theorem 82119
- Theorems of Camillo and Stephenson 82119
- The Basic Ring and Module of a Semiperfect Ring 83120
- The Regularity Condition and Small's Theorem 83120
- Reduced Rank 84121
- Finitely Embedded Rings and Modules: Theorems of Vámos and Beachy 84121
- The Endomorphism Ring of Noetherian and Artinian Modules 85122
- Fitting's Lemma 86123
- Köthe-Levitzki Theorem 87124
- Levitzki-Fitting Theorem 88125
- Kolchin's Theorem 89126
- Historical Notes on Local and Semilocal Rings 90127
- Further Notes for Chapter 3 92129
- Free Subgroups of GL(n,F) 92129
- Sanov's Theorem 92129
- Hartley-Pickel Theorem 93130
- Steinitz and Semi-Steinitz Rings 93130
- Free Direct Summands 93130
- Essentially Nilpotent Ideals 93130
- Comment on the Köthe Radical 94131

- Chapter 4. Direct Product Decompositions of von Neumann Regular Rings and Self-injective Rings 95132
- Clean Rings 96133
- Flat Modules 96133
- Character Modules and the Bourbaki-Lambek Theorem 97134
- When Everybody Is Flat 97134
- Singular Splitting 98135
- Utumi's Theorems 99136
- Weak or F.G Injectivity 100137
- Abelian VNR Rings 100137
- The Maximal Regular Ideal 101138
- Products of Matrix Rings over Abelian VNR Rings 101138
- Products of Full Linear Rings 102139
- Dedekind Finite 102139
- Jacobson's Theorem 102139
- Shepherdson's and Montgomery's Examples 103140
- Group Algebras in Characteristic 0 Are Dedekind Finite 103140

#### Readership

Graduate students, research mathematicians, and other scientists interested in the history of mathematics and science.

#### Reviews

[Regarding Chapter 18,

-- The Times of Trenton

This book offers a well-written and very detailed survey of a century of ring theory, module theory and, more generally, associative algebra. The author has selected a great many topics within this ambit and has done an excellent job in presenting them.

-- Mathematical Reviews