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The text of the first volume of the book covers the major topics in ring and module theory and includes both fundamental classical results and more recent developments. The basic tools of investigation are methods from the theory of modules, which allow a very simple and clear approach both to classical and new results. An unusual main feature of this book is the use of the technique of quivers for studying the structure of rings. A considerable part of the first volume of the book is devoted to a study of special classes of rings and algebras, such as serial rings, hereditary rings, semidistributive rings and tiled orders. Many results of this text until now have been available in journal articles only.

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Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated?

The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.

The algorithms to answer questions such as those posed above are an important part of algebraic geometry. Although the algorithmic roots of algebraic geometry are old, it is only in the last forty years that computational methods have regained their earlier prominence. New algorithms, coupled with the power of fast computers, have led to both theoretical advances and interesting applications, for example in robotics and in geometric theorem proving.

In addition to enhancing the text of the second edition, with over 200 pages reflecting changes to enhance clarity and correctness, this third edition of Ideals, Varieties and Algorithms includes: A significantly updated section on Maple in Appendix C; Updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica and SINGULAR; A shorter proof of the Extension Theorem presented in Section 6 of Chapter 3.

From the 2^nd Edition:

"I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures, and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry." -The American Mathematical Monthly

Customer Reviews (4)

too bad
the book is not new, and online it's write "brand new", the book is used buy someone else already. and I also think the book is like a copy version.

Good book (for Math people)
I actually haven't gone through the entire book, but the parts that I did go through were interesting. The book was (I thought) a little confusing to understand, but that may just be because it is for higher level abstract math. Lost of proofs and interesting things if your interested in abstract math!

Very clearly written.
This book explains the concepts well and the proofs are written carefully and clearly. This is a math book, so you should expect the usual Definition-Theorem-Proof format, but the exposition is progressive, and the material is very well motivated. I had to implement an algorithm for the computation of Groebner bases, and I borrowed this book from the University library to learn about the Buchberger algorithm and related topics, and thanks to this book I was able to understand this algorithm and its correctness proof without the help of anyone. Of course Buchberger's algorithm is only a small part of this book, and you can find a lot of important topics in it.
Finally, note that the second part of Madhu Sudan's Algebra and Computation course at MIT follows this book closely. So at least one top expert in the field seems to have a good opinion of the book.

Careful about production error
This is the 3rd edition of a popular reference on the subject. There was some production error with earlier version of the book. Even with the latest version, the authors have provided 14-page worth of correctionsfor the 1st printing of the 3rd edition. See www.cs.amherst.edu/~dac/iva/3ed1.pdf.
So buyers may be better off waiting for a corrected later version from the publisher. ... Read more

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This book is an expanded text for a graduate course in commutative algebra, focusing on the algebraic underpinnings of algebraic geometry and of number theory. Accordingly, the theory of affine algebras is featured, treated both directly and via the theory of Noetherian and Artinian modules, and the theory of graded algebras is included to provide the foundation for projective varieties. Major topics include the theory of modules over a principal ideal domain, and its applications to matrix theory (including the Jordan decomposition), the Galois theory of field extensions, transcendence degree, the prime spectrum of an algebra, localization, and the classical theory of Noetherian and Artinian rings. Later chapters include some algebraic theory of elliptic curves (featuring the Mordell-Weil theorem) and valuation theory, including local fields.

One feature of the book is an extension of the text through a series of appendices. This permits the inclusion of more advanced material, such as transcendental field extensions, the discriminant and resultant, the theory of Dedekind domains, and basic theorems of rings of algebraic integers. An extended appendix on derivations includes the Jacobian conjecture and Makar-Limanov's theory of locally nilpotent derivations. Gröbner bases can be found in another appendix.

Exercises provide a further extension of the text. The book can be used both as a textbook and as a reference source. ... Read more

Customer Reviews (1)

Covers recently developed subjects.
This textbook is meant as a sequel to introductory-ones of algebra.
There are already good textbooks by Cohn and Jacobson, say. I beleive this one nicely covers the subjects developed after the publishing of said ones, hence serves as a companion rather than competing with them.
... Read more

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The central subject of the book - the theory of shift-invariant algebras - is an outgrowth of the established theory of generalized analytic functions. Associated subalgebras of almost periodic functions of real variables and of bounded analytic functions on the unit disc are carried along within the general framework. In particular, it is shown that the algebra of almost periodic functions with spectrum in a semigroup of the reals does not have a half-plane-corona if and only if all non-negative semicharacters of the semigroup are monotone decreasing, or equivalently, if and only if the strong hull of the semigroup coincides with the positive half of its group envelope.  Under the same conditions the corresponding subalgebra of bounded analytic functions on the disc has neither a half-plane-corona nor a disc-corona. There are given characterizations of semigroups such that classical theorems of complex analysis hold on the associated shift-invariant algebras. Bourgain algebras, orthogonal measures, and primary ideals of big disc algebras are described. The notion of a harmonic function is extended on compact abelian groups, and corresponding Fatou-type theorems are proven. Important classes of inductive limits of standard uniform algebras, including Blasche algebras, are introduced and studied. In particular, it is shown that algebras of hyper-analytic functions, associated with families of inner functions, do not have a big-disc-corona.  ... Read more

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Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields. The first stage of this development of the theory is associated with its founder, J.F. Ritt (1893-1951), and R. Cohn, whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrown the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations with delay, algebraic structures with operators, group and semigroup rings.

The monograph is intended for graduate students and researchers in difference and differential algebra, commutative algebra, ring theory, and algebraic geometry. The book is self-contained; it requires no prerequisites other than the knowledge of basic algebraic concepts and a mathematical maturity of an advanced undergraduate.

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Addressed to the researchers working in the theory of table algebras and association schemes. Summarizes the research which was done at Bar-Ilan University in the academic year 1998/99. Softcover. ... Read more

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This book is the first monograph on the theory of endomorphism rings of Abelian groups. The theory is a rapidly developing area of algebra and has its origin in the theory of operators of vector spaves. The text contains additional information on groups themselves, introducing new concepts, methods, and classes of groups. All the main fields of the theory of endomorphism rings of Abelian groups from early results to the most recent are covered. Neighbouring results on endomorphism rings of modules are also mentioned. ... Read more

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This text offers comprehensive coverage of group cohomology, from introductory material through the most recent developments in the field. The primary motivation for this book is the interaction of group cohomology with representation theory, especially the geometry of support varieties over cohomology rings. The appendices, comprising computer calculations of the mod-2 cohomology rings of the groups whose orders divide 64, provide information useful for further developments in the field. A unique feature of this text is that it includes the concepts that are the subject of the calculations and are the source of some of the motivating conjectures for the computations. The programs for computing the cohomology rings were executed in the MAGMA computer algebra language. The text is a valuable resource for researchers in group cohomology and related disciplines. In addition, the book could be used as the text for an advanced graduate class or a graduate seminar. ... Read more

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This book covers the beautiful theory of resolutions of surface singularities in characteristic zero. The primary goal is to present in detail, and for the first time in one volume, two proofs for the existence of such resolutions. One construction was introduced by H.W.E. Jung, and another is due to O. Zariski. Jung's approach uses quasi-ordinary singularities and an explicit study of specific surfaces in affine three-space. In particular, a new proof of the Jung-Abhyankar theorem is given via ramification theory. Zariski's method, as presented, involves repeated normalisation and blowing up points. It also uses the uniformization of zero-dimensional valuations of function fields in two variables, for which a complete proof is given.

Despite the intention to serve graduate students and researchers of Commutative Algebra and Algebraic Geometry, a basic knowledge on these topics is necessary only. This is obtained by a thorough introduction of the needed algebraic tools in the two appendices.

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This new proceedings is a collection of papers presenting the recent developments and research in various fields of algebra, especially Lie Algebras, Rings and their related topics undertaken in the U.S.A., Russia, North Asia and Israel. Contributors include E. Zelmanov, the 1994 Fields Medalist, L. Bokut, Günter F. Pilz, Koichiro Harada, Alexander Kemer, V.K. Kharchenko, L. Makar-Limanov and Louis H. Rowen. ... Read more

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This is the first self-contained summary of non-Noetherianorders in a simple Artinian ring, a subject in which much progress hasbeen made in the last decade.
The contents of the book are mainly Dubrovin valuation rings andsemi-hereditary orders, including Prüfer and semi-local Bezoutorders, which are considered, in a sense, as global theories ofDubrovin valuation rings. These are then developed further, andapplied to give some examples such as Dubrovin valuation rings incrossed product algebras, semi-hereditary maximal order in certainmatrix rings, and the idealizers of semi-hereditary orders andHenselization of Bezout orders.
Audience: This volume will be of interest to researchers andgraduate students whose work involves non-commutative ring theory andmodule theory. ... Read more

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This volume has developed from courses given at Moscow StateUniversity. The main purpose of the material presented is to introducethe concepts, results and problems of contemporary algebra, assumingsome knowledge of the standard theory of linear algebra and vectorspaces. One important aspect is also to demonstrate how the conceptsdiscussed relate to each other and how they work in practice. The bookbegins with an introduction to the fundamental concepts of groups,rings, fields and modules and their representations. The sevenchapters which follow are devoted respectively to the followingtopics: commutative algebra; groups; associative rings; Lie algebras;homological algebra; algebraic groups; and varieties of algebras. Thevolume concludes with a supplement dealing with set theory, referencesand indices.
The book is as self-contained as possible.
For graduate students and researchers wishing to obtain a goodintroduction to the concepts of contemporary algebra.
... Read more

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Based on invited lectures at the 1992 Canadian Algebra Seminar,this volume represents an up-to-date and unique report onfinite-dimensional algebras as a subject with many seriousinteractions with other mathematical disciplines,including algebraicgroups and Lie theory, automorphic forms, sheaf theory, finite groups,and homological algebra. It will interest mathematicians and graduatestudents in these and related subjects as an introduction to researchin an area of increasing relevance and importance.
... Read more

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During the last few decades the ideas, methods, and results ofthe theory of Boolean algebras have played an increasing role invarious branches of mathematics and cybernetics.
This monograph is devoted to the fundamentals of the theory of Booleanconstructions in universal algebra. Also considered are the problemsof presenting different varieties of universal algebra with theseconstructions, and applications for investigating the spectra andskeletons of varieties of universal algebras.
For researchers whose work involves universal algebra and logic.
... Read more

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This volume consists of twenty-one articles by many of the mostprominent researchers in non-Noetherian commutative ring theory. Thearticles combine in various degrees surveys of past results, recentresults that have never before seen print, open problems, and anextensive bibliography. One hundred open problems supplied by theauthors have been collected in the volume's concluding chapter. Theentire collection provides a comprehensive survey of the developmentof the field over the last ten years and points to future directionsof research in the area.
Audience: Researchers and graduate students; the volume is anappropriate source of material for several semester-longgraduate-level seminars and courses. ... Read more

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This outstanding reference presents the latest advances in zero-dimensional commutative rings and commutative algebra--illustrating the research frontier with 52 open problems together with comments on the relevant literature and offering a comprehensive index permitting easy access to information. ... Read more