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The selected contributions in this volume originated atthe Sundance conference, which was devoted to discussions of currentwork in the area of free resolutions.The papers include newresearch, not otherwise published, and expositions that developcurrent problems likely to influence future developments in the field. ... Read more

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The theory of generalized inverses of real or complex matrices is a well-developed and well-documented subject.However, the wider subject of generalized inverses of matrices over rings has reached a state for a comprehensive treatment only recently.The author, who contributed to this development, provides a book for students of the subject.Mathematicians working in g-inverses of matrices, algebraists and control theorists will be interested in the results presented here.The book would also be suitable for graduate courses on G-inverses in algebra. ... Read more

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This book studies the universal constructions and properties in categories of commutative algebras, bringing out the specific properties that make commutative algebra and algebraic geometry work. Two universal constructions are presented and used here for the first time. The author shows that the concepts and constructions arising in commutative algebra and algebraic geometry are not bound so tightly to the absolute universe of rings, but possess a universality that is independent of them and can be interpreted in various categories of discourse. This brings new flexibility to classical commutative algebra and affords the possibility of extending the domain of validity and the application of the vast number of results obtained in classical commutative algebra. This innovative and original work will interest mathematicians in a range of specialities, including algebraists, categoricians, and algebraic geometers. ... Read more

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The already broad range of applications of ring theory has been enhanced in the eighties by the increasing interest in algebraic structures of considerable complexity, the so-called class of quantum groups. One of the fundamental properties of quantum groups is that they are modelled by associative coordinate rings possessing a canonical basis, which allows for the use of algorithmic structures based on Groebner bases to study them. This book develops these methods in a self-contained way, concentrating on an in-depth study of the notion of a vast class of non-commutative rings (encompassing most quantum groups), the so-called Poincaré-Birkhoff-Witt rings. We include algorithms which treat essential aspects like ideals and (bi)modules, the calculation of homological dimension and of the Gelfand-Kirillov dimension, the Hilbert-Samuel polynomial, primality tests for prime ideals, etc. ... Read more

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Provides an elementary introduction to these subjects. Author gives additive number theoretic results like derivation of LaGrange's foursquare theorem from Minkowski's result in the geometry of numbers. ... Read more

Customer Reviews (1)

Poor Quality Production
This book is a fair short treatment of algebra and number theory at the graduate level. I am not writing a review of its pedagogical or mathematical merits. My problem with the book is its sells for about [$$$] bucks for 153 pages but the quality of printing is so poor it makes the book hard to read. The pages are not opaque which means print from other pages appears through. The line spacing is narrow and the inner margins are very small. The hard cover seems to me to be of inferior quality as well. ... Read more

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The central theme of this volume is commutative algebra, with emphasis on special graded algebras, which are increasingly of interest in problems of algebraic geometry, combinatorics and computer algebra. Most of the papers have partly survey character, but are research-oriented, aiming at classification and structural results. ... Read more

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Computational methods are an established tool in algebraic geometry and commutative algebra, the key element being the theory of Gröbner bases. This book represents the state of the art in computational algebraic geometry and encapsulates many of the most interesting trends and developments in the field. There are two articles on open problems, orienting the reader to the subject's direction, four surveys describing the most interesting work, and four original research papers. There is also an introduction to the theory of Gröbner bases and their use in computation. Though the perspective of the book is mathematical, it does relate the abstract and the experimental tendencies in the field. Consequently, it will appeal to computer scientists interested in symbolic computation, robotics or Gröbner bases, as well as mathematicians interested in algebraic geometry, commutative algebra, or the classification of algebras. ... Read more

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Group rings play a central role in the theory ofrepresentations of groups and are very interesting algebraic objectsin their own right. In their study, many branches of algebra come to arich interplay. This book takes the reader from beginning to researchlevel and contains many topics that, so far, were only found in paperspublished in scientific journals and, whenever possible, offers newproofs of known results. It also includes many historical notes andsome applications. Audience: This book will be of interest to mathematiciansworking in the area of group rings and it serves as an introduction ofthe subject to graduate students. ... Read more

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Devoted to the theory of Lie algebras and algebraic groups, this book includes a large amount of commutative algebra and algebraic geometry so as to make it as self-contained as possible. The aim of the book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included, and some recent results are discussed in the final chapters.

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This book has a nonstandard choice of topics, including material on differential galois groups and proofs of the transcendence of e and pi. 

The author uses a conversational tone and has included a selection of stamps to accompany the text.

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This book will be particularly valuable to the American student because it covers material that is not available in any other textbooks or monographs. The subject of the book is not restricted to commutative algebra developed as a pure discipline for its own sake, nor is it aimed only at algebraic geometry where the intrinsic geometry of a general n-dimensional variety plays the central role. Instead, this book is developed around the vital theme that certain areas of both subjects are best understood together. This link between the two subjects, forged in the nineteenth century, built further by Krull and Zariski, remains as active as ever. In this book, the reader will find as the same time a leisurely and clear exposition of the basic definitions and results in both algebra and geometry, as well as an cxposition of the important recent progress fue to Quillen-Suslin, Evans-Eisenbud, Szpiro, Mohan Kumar and others. The ample exercises are another excellent feature. Professor Kunz has filled a longstanding need for an introduction to commutative algebra and algebraic geometry that emphasizes the concrete elementary nature of objects with which both subjects began. ... Read more

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This book provides a comprehensive treatment of Gröbner bases theory embedded in an introduction to commutative algebra from a computational point of view. The centerpiece of Gröbner bases theory is the Buchberger algorithm, which provides a common generalization of the Euclidean algorithm and the Gaussian elimination algorithm to multivariate polynomial rings. The book explains how the Buchberger algorithm and the theory surrounding it are eminently important both for the mathematical theory and for computational applications. A number of results such as optimized version of the Buchberger algorithm are presented in textbook format for the first time.

This book requires no prerequisites other than the mathematical maturity of an advanced undergraduate and is therefore well suited for use as a textbook. At the same time, the comprehensive treatment makes it a valuable source of reference on Gröbner bases theory for mathematicians, computer scientists, and others. Placing a strong emphasis on algorithms and their verification, while making no sacrifices in mathematical rigor, the book spans a bridge between mathematics and computer science. ... Read more

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Quadratic Algebras, Clifford Algebras, and Arithmetic Forms introduces mathematicians to the large and dynamic area of algebras and forms over commutative rings. The book begins very elementary and progresses gradually in its degree of difficulty. Topics include the connection between quadratic algebras, Clifford algebras and quadratic forms, Brauer groups, the matrix theory of Clifford algebras over fields, Witt groups of quadratic and symmetric bilinear forms. Some of the new results included by the author concern the representation of Clifford algebras, the structure of Arf algebra in the free case, connections between the group of isomorphic classes of finitely generated projectives of rank one and arithmetic results about the quadratic Witt group. ... Read more

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Algebra II is a two-part survey on the subject of non-commutative rings and algebras, with the second part focused on the theory of identities of these and other algebraic systems. It provides a broad overview of the most modern trends encountered in non-commutative algebra, as well as the numerous connections between algebraic theories and other areas of mathematics. A wide selection of examples of non-commutative rings is given at the start. Throughout the book, the authors include the historical background of the trends they are discussing. The authors, who are among the most prominent Soviet algebraists, share with their readers their knowledge of the subject, giving them a unique opportunity to learn the material from mathematicians who have made major contributions to it. This is especially true in the case of the theory of identities in varieties of algebraic objects where Soviet mathematicians have been a moving force behind this approach. This monograph on associative rings and algebras, group theory and algebraic geometry is intended for researchers and students. ... Read more

Customer Reviews (2)

Recreational but no-nonsense.
An off-hand account of algebra by one of the best authorities of thesubject. Recommended as a "serious" pass-time.

Modern Mathematics from the viewpoint of Algebra.
The author explains not just the definitions but also the 'philosophy' of algebra with highly non- trivial examples from geometry,analysis & topology not to mention algebra itself. Examples motivate the concepts and show their usefulness. There islittle number of formal proofs, but theorems areusually supported by natural 'explanations' whichmay be even more adequate. Frequently a single example in this book can replace a substantial theory of the corresponding subject. For example, about axioms of projective geometry. A reader with no knowledge of modern algebra cantry reading it ,but it would be more interestingif familiar with subjects of undergraduate mathe matics. Then an undergraduate reader will get some ideas how modern math uses sophiscated tools from algebra to understand other subjects. Another merit of this book is that it can besmoothly followed by the author's famous 2-volume textbook on algebraic geometry. ... Read more

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This volume, a tribute to the work of Robert Gilmer, consists of twenty-four articles authored by his most prominent students and followers. These articles combine surveys of past work by Gilmer and others, recent results which have never before seen print, open problems, and extensive bibliographies. The entire collection provides an in-depth overview of the topics of research in a significant and large area of commutative algebra.

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In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra. ... Read more