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

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.

Table of Contents:
Part I: An Array of Twentieth Century Associative Algebra; Direct product and sums of rings and modules and the structure of fields; Introduction to ring theory: Schur's lemma and semisimple rings, prime and primitive rings, nil, prime and Jacobson radicals; Direct sum decompositions of projective and injective modules; Direct product decompositions of von Neumann regular rings and self-injective rings; Direct sums of cyclic modules; When injectives are flat: Coherent FP-injective rings; Direct decompositions and dual generalizations of Noetherian rings; Completely decomposable modules and the Krull-Schmidt-Azumaya theorem; Polynomial rings over Vamosian and Kerr rings, valuation rings and Prufer rings; Isomorphic polynomial rings and matrix rings; Group rings and Maschke's theorem revisited; Maximal quotient rings; Morita duality and dual rings; Krull and global dimensions; Polynomial identities and PI-rings; Unions of primes, prime avoidance, associated prime ideals, ACC on irreducible ideals and annihilator ideals in commutative rings; Dedekind's theorem on the independence of automorphisms revisited; Part II: Snapshots of Some Mathematical Friends and Places; Snapshots of some mathematical friends and places; Envoi to my century; Bibliography; Register of names; Index of terms and authors of theorems