This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

The chemistry of copper is extremely rich because it can easily access Cu0, CuI, CuII, and CuIII oxidation states allowing it to act through one-electron or two-electron processes. As a result, both radical pathways and powerful two-electron bond forming pathways via organmetallic intermediates, similar to those of palladium, can occur. In addition, the different oxidation states of copper associate well with a large number of different functional groups via Lewis acid interactions or π-coordination. In total, these feature confer a remarkably broad range of activities allowing copper to catalyze the oxidation and oxidative union of many substrates.

Oxygen is a highly atom economical, environmentally benign, and abundant oxidant, which makes it ideal in many ways.1 The high activation energies in the reactions of oxygen require that catalysts be employed.2 In combination with molecular oxygen, the chemistry of copper catalysis increases exponentially since oxygen can act as either a sink for electrons (oxidase activity) and/or as a source of oxygen atoms that are incorporated into the product (oxygenase activity). The oxidation of copper with oxygen is a facile process allowing catalytic turnover in net oxidative processes and ready access to the higher CuIII oxidation state, which enables a range of powerful transformations including two-electron reductive elimination to CuI. Molecular oxygen is also not hampered by toxic byproducts, being either reduced to water, occasionally via H2O2 (oxidase activity) or incorporated into the target structure with high atom economy (oxygenase activity). Such oxidations using oxygen or air (21% oxygen) have been employed safely in numerous commodity chemical continuous and batch processes.3

However, batch reactors employing volatile hydrocarbon solvents require that oxygen concentrations be kept low in the head space (typically <5–11%) to avoid flammable mixtures, which can limit the oxygen concentration in the reaction mixture.4,5,6 A number of alternate approaches have been developed allowing oxidation chemistry to be used safely across a broader array of conditions. For example, use of carbon dioxide instead of nitrogen as a diluent leads to reduced flammability.5 Alternately, water can be added to moderate the flammability allowing even pure oxygen to be employed.6 New reactor designs also allow pure oxygen to be used instead of diluted oxygen by maintaining gas bubbles in the solvent, which greatly improves reaction rates and prevents the build up of higher concentrations of oxygen in the head space.4a,7 Supercritical carbon dioxide has been found to be advantageous as a solvent due its chemical inertness towards oxidizing agents and its complete miscibility with oxygen or air over a wide range of temperatures.8 An number of flow technologies9 including flow reactors,10 capillary flow reactors,11 microchannel/microstructure structure reactors,12 and membrane reactors13 limit the amount of or afford separation of hydrocarbon/oxygen vapor phase thereby reducing the potential for explosions.

Enzymatic oxidizing systems based upon copper that exploit the many advantages and unique aspects of copper as a catalyst and oxygen as an oxidant as described in the preceding paragraphs are well known. They represent a powerful set of catalysts able to direct beautiful redox chemistry in a highly site-selective and stereoselective manner on simple as well as highly functionalized molecules. This ability has inspired organic chemists to discover small molecule catalysts that can emulate such processes. In addition, copper has been recognized as a powerful catalyst in several industrial processes (e.g. phenol polymerization, Glaser-Hay alkyne coupling) stimulating the study of the fundamental reaction steps and the organometallic copper intermediates. These studies have inspiried the development of nonenzymatic copper catalysts. For these reasons, the study of copper catalysis using molecular oxygen has undergone explosive growth, from 30 citations per year in the 1980s to over 300 citations per year in the 2000s.

A number of elegant reviews on the subject of catalytic copper oxidation chemistry have appeared. Most recently, reviews provide selected coverage of copper catalysts14 or a discussion of their use in the aerobic functionalization of C–H bonds.15 Other recent reviews cover copper and other metal catalysts with a range of oxidants, including oxygen, but several reaction types are not covered.16 Several other works provide a valuable overview of earlier efforts in the field.17 This review comprehensively covers copper catalyzed oxidation chemistry using oxygen as the oxidant up through 2011. Stoichiometric reactions with copper are discussed, as necessary, to put the development of the catalytic processes in context. Mixed metal systems utilizing copper, such as palladium catalyzed Wacker processes, are not included here. Decomposition reactions involving copper/oxygen and model systems of copper enzymes are not discussed exhaustively. To facilitate analysis of the reactions under discussion, the current mechanistic hypothesis is provided for each reaction. As our understanding of the basic chemical steps involving copper improve, it is expected that many of these mechanisms will evolve accordingly.

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Aerobic Copper-Catalyzed Organic Reactions

This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups It aims to provide the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields Involutions are viewed as twisted forms of similarity classes of hermitian or bilinear forms, leading to new developments on the model of the algebraic theory of quadratic forms Besides classical groups, phenomena related to triality are also discussed, as well as groups of type F_4 or G_2 arising from exceptional Jordan or composition algebras Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type D_4 For research mathematicians and graduate students working in central simple algebras, algebraic groups, nonabelian Galois cohomology or Jordan algebras

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The Book of Involutions

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

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The Algebraic and Geometric Theory of Quadratic Forms

The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert, and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi–Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.

http://www.gbv.de/dms/goettingen/51399310X.pdf

Central Simple Algebras and Galois Cohomology

Division Algebras over Henselian Fields

Index Reduction Formulas for Twisted Flag Varieties, II

This monograph is the first book-length treatment of valuation theory on finite-dimensional division algebras, a subject of active and substantial research over the last forty years. Its development was spurred in the last decades of the twentieth century by important advances such as Amitsur's construction of noncrossed products and Platonov's solution of the Tannaka-Artin problem. This study is particularly timely because it approaches the subject from the perspective of associated graded structures. This new approach has been developed by the authors in the last few years and has significantly clarified the theory. Various constructions of division algebras are obtained as applications of the theory, such as noncrossed products and indecomposable algebras. In addition, the use of valuation theory in reduced Whitehead group calculations (after Hazrat and Wadsworth) and in essential dimension computations (after Baek and Merkurjev) is showcased. The intended audience consists of graduate students and research mathematicians.

Value Functions on Simple Algebras, and Associated Graded Rings

Valuation theory has classically meant the study of valuations on a commutative field. Such valuation theory has flourished for many decades, nourished by its connections with number theory and algebraic geometry. But there is also a noncommutative side of the subject in the study of valuations and valuation rings on division rings. This aspect has blossomed only in the last twenty odd years, and it is not so well known as its commutative counterpart. We give in this paper a survey of valuations and valuation theory for division rings finite dimensional over their centers. We describe the associated theory and also some of the most significant constructions that have been given as applications. The earliest use of valuations on noncommutative division rings, to my knowledge, was by Hasse in his work in [Ha] in 1931 on orders over central simple algebras over p-adic fields. In the ’40’s and ’50’s there was a little further work with valuations on division algebras over fields with complete discrete valuations. Also, there was some discussion of valuations on division algebras in Schilling’s work [Schi1], [Schi2], mostly observing that some results about valuations remain valid without assuming that the ambient field be commutative. One can speculate that there was little attention to valuations on division algebras because too often for the division algebras of interest there was no apparent valuation available. Additionally, it was not fully clear what to take as the definition of a valuation on a division ring, since the concepts of valuation and valuation ring are not equivalent in the noncommutative setting. It was not until the late 1970’s and the 1980’s that valuation theory on division algebras began to develop substantially. This was due in large measure to the realization that some of the major constructions of counterexamples in the 1970’s could best be understood using valuation theory. This applies to Amitsur’s construction of noncrossed product division algebras and also to the constructions of division algebras with nontrivial SK1 by Platonov and by Yanchevskĭi. It began to

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Valuation Theory on Finite Dimensional Division Algebras

Let D be a division algebra finite dimensional over its center F. It is shown that a (Krull) valuation v on F extends to a valuation on D iff v extends uniquely to each commutative field K with F c K c D. A valuation on a division ring is a function v: D* G (where G is a totally ordered group) satisfying, for all a, b E D* = D {O}, (i) v(ab) = v(a) + v(b); (ii) v(a + b) > min{v(a), v(b)} if b # -a. It is convenient to use additive notation for G, though G is not assumed to be abelian. If E is any subring of D, v restricts to a valuation on E, and we say that (D, v) is an extension of (E, v). The standard reference for valuations on division rings is Schilling's book [S]. One of the fundamental features of commutative valuation theory is that for fields (meaning commutative fields) F c K, every valuation on F has an extension to K (possibly many different extensions). However, if we replace the fields F and K by division rings this extension property does not always hold. This failure is a major obstacle in noncommutative valuation theory. On the positive side, it is well known that if F is a field with Henselian valuation v, then v extends to a valuation on any F-division algebra D finite dimensional over F. We generalize this Henselian result to give a precise criterion for when a valuation on a field F extends to one on a given finite dimensional F-central division algebra. This criterion was proved by P. M. Cohn in [C, Theorem 1] for valuations v on F with value group v(F*) a subgroup of the additive group of real numbers, i.e., when the valuation ring of (F, v) has Krull dimension 1. Cohn's argument used completions, and does not appear to generalize readily to higher Krull dimensions. However, there are significant examples of valued division algebras with rings of Krull dimension at least 2. Notably, such algebras have been used in the construction of noncrossed product algebras (cf. [A, T, JW]) and also in the examples of division algebras D with SK1(D) nontrivial (cf. [P, DK]). So, it is worthwhile to see that the extension criterion holds without any Krull dimension restriction. Received by the editors August 19, 1985. 1980 Mathematics Subject Classification. Primary 16A39; Secondary 12E15. 'Supported in part by the N.S.F. ?01986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

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Adrian R. Wadsworth

Papers

Division Algebras over Henselian Fields

Index Reduction Formulas for Twisted Flag Varieties, II

Value Functions on Simple Algebras, and Associated Graded Rings

Valuation Theory on Finite Dimensional Division Algebras

Extending valuations to finite dimensional division algebras