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Graduate Algebra: Commutative View
 
Louis Halle Rowen Bar-Ilan University, Ramat Gan, Israel
Graduate Algebra: Commutative View
eBook ISBN:  978-1-4704-1152-7
Product Code:  GSM/73.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Graduate Algebra: Commutative View
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Graduate Algebra: Commutative View
Louis Halle Rowen Bar-Ilan University, Ramat Gan, Israel
eBook ISBN:  978-1-4704-1152-7
Product Code:  GSM/73.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 732006; 438 pp
    MSC: 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 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.

    Readership

    Graduate 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
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 732006; 438 pp
MSC: 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 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.

Readership

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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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