[For Vol. I see the preceding review.]
From Introduction to Volume II: “After a brief treatment of those techniques of homology and cohomology needed for rings, the book turns specifically to certain types of rings that have been objects of intense investigation in recent years - Azumaya algebras (5.3), rings with polynomial identity (Chapter 6), division rings, and, more generally, (finite dimensional) central simple algebras (Chapter 7), group rings (8.1 and 8.2), and enveloping algebras (8.3)”.
Contents: Introduction to Volume II; Table of Principal Notation for Volume II; Chapter 5, Homology and Cohomology: 5.0 Preliminaries about Diagrams, 5.1 Resolutions and Projective and Injective Dimension, 5.2 Homology, Cohomology and Derived Functors, 5.3 Separable Algebras and Azumaya Algebras, Appendix to Chapter 5: The Grothendieck group, and Quillen’s Theorem Revisited, Exercises; Chapter 6, Rings with Polynomial Identities and Affine Algebras: 6.1 Rings with Polynomial Identities, 6.2 Affine Algebras, 6.3 Affine PI-Algebras, 6.4 Relatively Free PI-Rings and T-Ideals, Exercises; Chapter 7, Central Simple Algebras: 7.1 Structure of Central Simple Algebras, 7.2 The Brauer Group and the Merkur’ev-Suslin Theorem, 7.3 Special Results, Appendix A: The Arithmetic Theory and the Brauer Group, Appendix B: The Merkur’ev-Suslin Theorem and K-Theory, Appendix C: Generics, Exercises; Chapter 8, Rings from Representation Theory: 8.1 General Structure Theory of Group Algebras, 8.2 Noetherian Group Rings, 8.3 Enveloping Algebras, 8.4 General Ring Theoretic Methods, Appendix to Chapter 8: Moody’s Theorem, Exercises; Dimensions for Modules and Rings; Major Theorems and Counterexamples for Volume II; References; Index of References According to Section; Cumulative Subject Index for Volumes I and II.
In Chapter 5 the (left) global dimension of a ring R (gl.dim R) is determined in the usual way, and the connection between gl.dim R and that of some rings related to R is established. For any ring R the group \(K_ 0(R)\) is defined and theorems by Serre and Quillen on the structure of \(K_ 0(R)\) for a graded ring with gl.dim R\(<\infty\) are proven. Another well known ring result proved in this chapter is the famous theorem by Suslin and Quillen, the positive solution of Serre’s conjecture on projective modules over polynomial rings. The Chapter contains also the usual elementary material on homology and cohomology (definition of appropriate groups, long exact sequences, derived functors, functors Tor and Ext). The developed theory is used for investigation of separable algebras and Azumaya algebras. Hochschild’s cohomology groups are defined too and the cohomological version of Wedderburn’s main theorem is proven (MacLane in his “Homology” calls it “Whitehead-Hochschild’s” theorem). The Grothendieck group (for any abelian category) is determined, and Quillen’s theorem mentioned on \(K_ 0(R)\) for a graded ring R is formulated in terms of Grothendieck group of the category R- Mod.
Chapter 6 deals with PI-rings and affine algebras.
In Section 6.1 the theory of PI-rings is developed using two notions which are absent in the books by Jacobson, Herstein, Faith mentioned above: central polynomials and the PI-degree. In terms of PI-degree Kaplansky’s theorem on primitive PI-rings is formulated. The classical theorem of embedding a semiprime PI-ring into a matrix ring is proven. The theorem (called the Artin-Procesi Theorem) is proven which characterizes Azumaya algebras of constant rank in terms of multilinear identities of \(M_ n({\mathbb{Z}})\) and PI-degree. For any commutative ring C the algebra \(C_ n\{Y\}\) of generic \(n\times n\)-matrices is determined. \(C_ n\{Y\}\) turns out to be a free algebra in the variety of C-algebras satisfying all identities of \(M_ n(C[\Lambda])\), where \(\Lambda\) is a countable set of indeterminates. The algebra \(C_ n\{Y\}\) appears in many constructions in the book under review. Digression treats generalized identities and shows their applications to the theory of strongly primitive rings. In Supplement identities and generalized identities with involution are considered.
Section 6.2 deals with affine algebras, i.e. C-algebras which are finitely generated as algebras. After some preliminary considerations the author passes to main topics of the section, Kurosh’s problem, and the growth of algebras and the Gelfand-Kirillov dimension. The famous Golod- Shafarevich example is considered in full details. Other results on Kurosh’s problem are given in Exercises. The growth function of an algebra over a field is defined, and the notion of exponential, subexponential, polynomial bounded growth is introduced. The meaning of such notion is illustrated by the following two examples. Assume R is a domain not necessarily with 1 which does not contain a copy of the free algebra \(F\{X_ 1,X_ 2\}\) without 1 (this is the case if R has subexponential growth); then R is left and right Ore (Jategaonkar). Suppose R is an affine, semiprime F-algebra with ACC on left and right annihilators; if R has subexponential growth then R is Goldie (Irving- Small). Then the Gelfand-Kirillov dimension of an affine algebra R (GKdim(R)) is defined. The connection between GKdim(R) and Gelfand- Kirillov dimension of some rings related to R is considered. A special attention is given to the question: which \(\alpha\) may be GKdim(R) for some R? The answer (based on results by Bergman and Borho-Kraft) sounds as follows: let \(\alpha =GK\dim (R)\) for R affine; there are the following possibilities only: \(\alpha =0\) iff R is finite dimensional over F; \(\alpha =1\) iff R is finitely generated over a central subring isomorphic to F[X]; otherwise \(\alpha\geq 2\), and any such \(\alpha\) can appear.
In Section 6.3 affine PI-algebras are considered. The Nullstellensatz for such algebras is discussed, in particular the Amitsur-Procesi theorem is proven which states: if R is an affine algebra over a commutative Jacobson ring C, then R is Jacobson satisf From the introduction.