The most extensive changes in this edition occur in the segment of the book devoted to commutative algebra, especially in Chapter 7, Commutative Ideal Theory: General Theory and Noetherian Rings; Chapter 8, Field Theory; and Chapter 9, Valuation Theory. In Chapter 7 we give an improved account of integral dependence, highlighting relations between a ring and its integral extensions (“lying over,” “going-up,” and “going-down” theorems). Section 7.7, Integrally Closed Domains, is new, as are three sections in Chapter 8: 8.13, Transcendency Bases for Domains; 8.18, Tensor Products of Fields; and 8.19, Free Composites of Fields. The latter two are taken from Volume III of our Lectures in Abstract Algebra (D. Van Nostrand 1964; Springer-Verlag, 1980). The most notable addition to Chapter 9 is Krasner’s lemma, used to give an improved proof of a classical theorem of Kurschak’s lemma (1913). We also give an improved proof of the theorem on extensions of absolute values to a finite dimensional extension of a field (Theorem 9.13) based on the concept of composite of a field considered in the new section 8.18.
In Chapter 4, Basic Structure Theory of Rings, we give improved accounts of the characterization of finite dimensional splitting fields of central simple algebras and of the fact that the Brauer classes of central simple algebras over a given field constitute a set—a fact which is needed to define the Brauer group Br(F). In the chapter on homological algebra (Chapter 6), we give an improved proof of the existence of a projective resolution of a short exact sequence of modules.
A number of new exercises have been added and some defective ones have been deleted.
Some of the changes we have made were inspired by suggestions made by our colleagues, Walter Feit, George Seligman, and Tsuneo Tamagawa. They, as well as Ronnie Lee, Sidney Porter (a former graduate student), and the Chinese translators of this book, Professors Cao Xi-hua and Wang Jian-pan, pointed out some errors in the first edition which are now corrected. We are indeed grateful for their interest and their important inputs to the new edition. Our heartfelt thanks are due also to F. D. Jacobson, for reading the proofs of this text and especially for updating the index.
January 1989
Nathan Jacobson