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Book DetailsGraduate Studies in MathematicsVolume: 233; 2023; 373 ppMSC: Primary 13; 11; 14
This book provides an introduction to classical methods in commutative algebra and their applications to number theory, algebraic geometry, and computational algebra. The use of number theory as a motivating theme throughout the book provides a rich and interesting context for the material covered. In addition, many results are reinterpreted from a geometric perspective, providing further insight and motivation for the study of commutative algebra.
The content covers the classical theory of Noetherian rings, including primary decomposition and dimension theory, topological methods such as completions, computational techniques, local methods and multiplicity theory, as well as some topics of a more arithmetic nature, including the theory of Dedekind rings, lattice embeddings, and Witt vectors. Homological methods appear in the author's sequel, Homological Methods in Commutative Algebra (Graduate Studies in Mathematics, Volume 234).
Overall, this book is an excellent resource for advanced undergraduates and beginning graduate students in algebra or number theory. It is also suitable for students in neighboring fields such as algebraic geometry who wish to develop a strong foundation in commutative algebra. Some parts of the book may be useful to supplement undergraduate courses in number theory, computational algebra or algebraic geometry. The clear and detailed presentation, the inclusion of computational techniques and arithmetic topics, and the numerous exercises make it a valuable addition to any library.
ReadershipGraduate students and researchers interested in commutative algebra.

Table of Contents

Chapters

Basics

Finiteness conditions

Factorization

Computational methods

Integral dependence

Lattice methods

Metric and topological methods

Geometric dictionary

Dimension theory

Local structure

Fields


Additional Material

Reviews

In my opinion, [this volume] is an excellent choice for a onesemester introductory course on commutative algebra.
Pramod Achar,Notices of the AMS 
Commutative algebra is at the crossroad between many fertile areas of mathematics. This book conveys the various points of view appropriately. It presents the connections with the most important applications of commutative algebra, such as number theory, algebraic geometry and computational algebra. The exercises vary from simple to hard, present many auxilary topics and treat important themes.
Ali Benhissi, zbMath


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
 Book Details
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This book provides an introduction to classical methods in commutative algebra and their applications to number theory, algebraic geometry, and computational algebra. The use of number theory as a motivating theme throughout the book provides a rich and interesting context for the material covered. In addition, many results are reinterpreted from a geometric perspective, providing further insight and motivation for the study of commutative algebra.
The content covers the classical theory of Noetherian rings, including primary decomposition and dimension theory, topological methods such as completions, computational techniques, local methods and multiplicity theory, as well as some topics of a more arithmetic nature, including the theory of Dedekind rings, lattice embeddings, and Witt vectors. Homological methods appear in the author's sequel, Homological Methods in Commutative Algebra (Graduate Studies in Mathematics, Volume 234).
Overall, this book is an excellent resource for advanced undergraduates and beginning graduate students in algebra or number theory. It is also suitable for students in neighboring fields such as algebraic geometry who wish to develop a strong foundation in commutative algebra. Some parts of the book may be useful to supplement undergraduate courses in number theory, computational algebra or algebraic geometry. The clear and detailed presentation, the inclusion of computational techniques and arithmetic topics, and the numerous exercises make it a valuable addition to any library.
Graduate students and researchers interested in commutative algebra.

Chapters

Basics

Finiteness conditions

Factorization

Computational methods

Integral dependence

Lattice methods

Metric and topological methods

Geometric dictionary

Dimension theory

Local structure

Fields

In my opinion, [this volume] is an excellent choice for a onesemester introductory course on commutative algebra.
Pramod Achar,Notices of the AMS 
Commutative algebra is at the crossroad between many fertile areas of mathematics. This book conveys the various points of view appropriately. It presents the connections with the most important applications of commutative algebra, such as number theory, algebraic geometry and computational algebra. The exercises vary from simple to hard, present many auxilary topics and treat important themes.
Ali Benhissi, zbMath