"Rings and Things and a Fine Array of Twentieth Century Associative Algebra" by 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.

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.


Contents
Symbols
Preface to the Second Edition
Acknowledgements to the Second Edition
Preface to the First Edition
Acknowledgements to the First Edition
Part I. An Array of Twentieth Century Associative Algebra
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 Priifer 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
Part II. Snapshots of Some Mathematical Friends and Places
18. Snapshots of Some Mathematical Friends and Places
Index to PartII (Snapshots)
Bibliography
Register of Names
Index of Terms and Authors of Theorems

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