Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Second Edition
About this Title
Carl Faith, Professor Emeritus, Rutgers University, New Brunswick, NJ
Publication: Mathematical Surveys and Monographs
Publication Year 1999: Volume 65
ISBNs: 978-0-8218-3672-9 (print); 978-1-4704-1292-0 (online)
MathSciNet review: MR1657671
MSC: Primary 01A60; Secondary 01A70, 13-03, 16-00, 16-03
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, Algebra: Rings, Modules and Categories, I and II (Springer-Verlag, 1973), are devoted to the development of modern associative algebra and ring and module theory. Those works serve as a foundation for the present survey, which includes a bibliography of over 1,600 references and is exhaustively indexed.
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.
Comments on the first edition:
“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
Graduate students, research mathematicians, and other scientists interested in the history of mathematics and science.
Table of Contents
- 1. Direct product and sums of rings and modules and the structure of fields
- 2. Introduction to ring theory: Schur’s Lemma and semisimple rings, prime and primitive rings, Noetherian and Artinian modules, nil, prime and Jacobson radicals
- 3. Direct decompositions of projective and injective modules
- 4. Direct product decompositions of von Neumann regular rings and self-injective rings
- 5. Direct sums of cyclic modules
- 6. When injectives are flat: Coherent FP-injective rings
- 7. Direct decompositions and dual generalizations of Noetherian rings
- 8. Completely decomposable modules and the Krull-Schmidt-Azumaya theorem
- 9. Polynomial rings over Vamosian and Kerr rings, valuation rings and Prüfer rings
- 10. Isomorphic polynomial rings and matrix rings
- 11. Group rings and Maschke’s theorem revisited
- 12. Maximal quotient rings
- 13. Morita duality and dual rings
- 14. Krull and global dimensions
- 15. Polynomial identities and PI-rings
- 16. Unions of primes, prime avoidance, associated prime ideals, ACC on irreducible ideals, and Annihilator ideals in commutative rings
- 17. Dedekind’s theorem on the independence of automorphisms revisited
- 18. Snapshots of some mathematical friends and places