Ring Theory: Volume I

[from Foreword] The theory of rings originally developed as a collection of brilliant insights for proving and improving diverse theorems in algebra, by extracting the essence of the theorems divested of superfluous hypotheses. This approach was startlingly successful and led to the search for solutions from “first principles”; moreover, there was a unifying structure theory due principally to Jacobson, and his book Structure of Rings (1956, rev. 1964) remains one of the principal references in the literature. (The other classic references are Herstein’s Carus Monograph Noncommutative Rings and, for the homological point of view, Cartan- Eilenberg’s book Homological Algebra.) However, as more and more “elementary” proofs found their way into the folklore, the structure of rings began to resemble a tower of Babel, where ring theorists speak such diverse languages (ring theory, module theory, category theory, etc.) that at times we cannot communicate effectively with one another. …

The principal object of these two volumes is to present the mainstream of ring theory, being a source both of results and of proofs. The first volume deals with general structural results and could serve as an introductory text; the second focuses on the classes of rings that have occupied most attention in the literature.

On the other hand, efforts were made to hold the material to readable size, leading to the following hierarchy: Main Text: • Supplementary material • Appendices • Exercises • Digressions