Electronic ISBN:  9781470411527 
Product Code:  GSM/73.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $56.80 

Book DetailsGraduate Studies in MathematicsVolume: 73; 2006; 438 ppMSC: Primary 28; Secondary 26; 31; 42; 46; 49; 81;
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 MordellWeil 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 MakarLimanov'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.ReadershipGraduate students interested in algebra, geometry, and number theory. Research mathematicians interested in algebra.

Table of Contents

Chapters

Chapter 0. Introduction and prerequisites

Exercises—Chapter 0

Part I. Modules

Chapter 1. Introduction to modules and their structure theory

Chapter 2. Finitely generated modules

Chapter 3. Simple modules and composition series

Exercises—Part I

Part II. Affine algebras and Noetherian rings

Introduction to Part II

Chapter 4. Galois theory of fields

Chapter 5. Algebras and affine fields

Chapter 6. Transcendence degree and the Krull dimension of a ring

Chapter 7. Modules and rings satisfying chain conditions

Chapter 8. Localization and the prime spectrum

Chapter 9. The Krull dimension theory of commutative Noetherian rings

Exercises—Part II

Part III. Applications to geometry and number theory

Introduction to Part III

Chapter 10. The algebraic foundations of geometry

Chapter 11. Applications to algebraic geometry over the rationals—Diophantine equations and elliptic curves

Chapter 12. Absolute values and valuation rings

Exercises—Part III


Additional Material

Reviews

The book is consistently organized in several layers (main text, supplements, appendices), which makes it a valuable source for readers of various levels, from graduate students to researchers.
European Mathematical Society Newsletter 
Its outstanding features are of many kinds, ranging from its underlying philosophy of keeping the material as much to the point as possible, thereby being highly efficient, up to the vast amount of topical extras that are barely found somewhere else. Already this first volume must be seen as a didactic masterpiece, profitable for both students and teachers likewise.
Zentralblatt MATH 
The author chose a very nice way to present all the basics needed to have a "solid basis in algebra", walking "direct to the goal" (my words), and showing the most important results of commutative algebra as soon and as simply as possible.
Mathematical Reviews


RequestsReview Copy – for reviewers who would like to review an AMS bookDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a coursePermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
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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 MordellWeil 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 MakarLimanov'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.
Graduate students interested in algebra, geometry, and number theory. Research mathematicians interested in algebra.

Chapters

Chapter 0. Introduction and prerequisites

Exercises—Chapter 0

Part I. Modules

Chapter 1. Introduction to modules and their structure theory

Chapter 2. Finitely generated modules

Chapter 3. Simple modules and composition series

Exercises—Part I

Part II. Affine algebras and Noetherian rings

Introduction to Part II

Chapter 4. Galois theory of fields

Chapter 5. Algebras and affine fields

Chapter 6. Transcendence degree and the Krull dimension of a ring

Chapter 7. Modules and rings satisfying chain conditions

Chapter 8. Localization and the prime spectrum

Chapter 9. The Krull dimension theory of commutative Noetherian rings

Exercises—Part II

Part III. Applications to geometry and number theory

Introduction to Part III

Chapter 10. The algebraic foundations of geometry

Chapter 11. Applications to algebraic geometry over the rationals—Diophantine equations and elliptic curves

Chapter 12. Absolute values and valuation rings

Exercises—Part III

The book is consistently organized in several layers (main text, supplements, appendices), which makes it a valuable source for readers of various levels, from graduate students to researchers.
European Mathematical Society Newsletter 
Its outstanding features are of many kinds, ranging from its underlying philosophy of keeping the material as much to the point as possible, thereby being highly efficient, up to the vast amount of topical extras that are barely found somewhere else. Already this first volume must be seen as a didactic masterpiece, profitable for both students and teachers likewise.
Zentralblatt MATH 
The author chose a very nice way to present all the basics needed to have a "solid basis in algebra", walking "direct to the goal" (my words), and showing the most important results of commutative algebra as soon and as simply as possible.
Mathematical Reviews