Rings and Things and a Fine Array of Twentieth Century Associative Algebra

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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. In the second part of the book, the author gives descriptive impressions of the last half of the twentieth century. 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 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.

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Editorial Reviews

Surveys over 125 years of associative algebras, focusing on ring and module theory. Topics of discussion in part one include categorical properties from theorems of Frobenius and Stickelberger on the primary decomposition of finite Abelian groups; Hilbert's basis theorem and his Nullstellensatz; the modern formulations of Hillbert's theorem by Krull and Goldman; Maschke's theorem on the representation theory of finite groups over a field; the fundamental theorems of Wedderburn on the structure of finite dimensional algebras and finite skew fields; and their extensions by Braver, Kaplansky, Chevally, and Goldie. Also included is a study of rings with chain condition on annihilator ideals pioneered by Noether, Artin, and Jacobson and refined and extended by many later mathematicians. Part two contains Faith's (professor emeritus, Rutgers U.) impressions of some mathematical friends and places from the last half of the 20th century. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

Table of Contents

Ch. 1 Direct Product and Sums of Rings and Modules and the Structure of Fields 1
Ch. 2 Introduction to Ring Theory: Schur's Lemma and Semisimple Rings, Prime and Primitive Rings, Noetherian and Artinian Modules, Nil, Prime and Jacobson Radicals 15
Ch. 3 Direct Decompositions of Projective and Injective Modules 51
Ch. 4 Direct Product Decompositions of von Neumann Regular Rings and Self-injective Rings 89
Ch. 5 Direct Sums of Cyclic Modules 105
Ch. 6 When Injectives Are Flat: Coherent FP-injective Rings 121
Ch. 7 Direct Decompositions and Dual Generalizations of Noetherian Rings 139
Ch. 8 Completely Decomposable Modules and the Krull-Schmidt-Azumaya Theorem 153
Ch. 9 Polynomial Rings over Vamosian and Kerr Rings, Valuation Rings and Prufer Rings 163
Ch. 10 Isomorphic Polynomial Rings and Matrix Rings 173
Ch. 11 Group Rings and Maschke's Theorem Revisited 175
Ch. 12 Maximal Quotient Rings 179
Ch. 13 Morita Duality and Dual Rings 193
Ch. 14 Krull and Global Dimensions 205
Ch. 15 Polynomials Identities and PI-Rings 221
Ch. 16 Unions of Primes, Prime Avoidance, Associated Prime Ideals, Acc on Irreducible Ideals, and Annihilator Ideals in Commutative Rings 229
Ch. 17 Dedekind's Theorem on the Independence of Automorphisms Revisited 243
Ch. 18 Some Mathematical Friends and Places 255
Bibliography 325
Register of Names 387
Index of Terms and Authors of Theorems 395
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