The book offers a comprehensive introduction to commutative algebra, featuring 400 solved exercises and a self-assessment test. It covers fundamental concepts, definitions, properties, and results, with a focus on multivariate polynomial rings and modules over a principal ideal domain (PID). The solutions provide essential details and aim for a well-balanced presentation. The book is suitable for advanced undergraduates or masters students with basic knowledge of finite fields, Abelian groups, and linear algebra, and encourages them to find their own proofs while providing detailed solutions to support their learning.

Format: Paperback / softback
Length: 392 pages
Publication date: 13 July 2024
Publisher: Springer International Publishing AG

Introduction

This book provides a first introduction to the fundamental concepts of commutative algebra. What sets it apart from other textbooks is the extensive collection of 400 solved exercises, providing readers with the opportunity to apply theoretical knowledge to practical problem solving, fostering a deeper and more thorough understanding of the subject. The topics presented here are not commonly found in a single text. Consequently, the first part presents definitions, properties, and results crucial for understanding and solving the exercises, serving also as a valuable reference. The second part contains the exercises and a section titled "True or False?" questions, which serves as a valid self-assessment test. Considerable effort has been invested in crafting solutions that provide the essential details, aiming for a well-balanced presentation. We intend to guide students systematically through the challenging process of writing mathematical proofs with formal correctness and clarity. Our approach is constructive, aiming to illustrate concepts by applying them to the analysis of multivariate polynomial rings and modules over a principal ideal domain (PID) whenever feasible. Algorithms for computing these objects facilitate the generation of diverse examples. In particular, the structure of finitely generated modules over a PID is analyzed using the Smith canonical form of matrices. Furthermore, various properties of polynomial rings are investigated through the application of Buchbergers Algorithm for computing Gröbner bases. This book is intended for advanced undergraduates or masters students, assuming only basic knowledge of finite fields, Abelian groups, and linear algebra. This approach aims to inspire the curiosity of readers and encourages them to find their own solutions to the exercises.

Contents

Part I: Definitions, Properties, and Results

Chapter 1: Introduction to Commutative Algebra

Chapter 2: Rings and Modules

Chapter 3: Ideals and Factorization

Chapter 4: Polynomial Rings

Chapter 5: Modules over a Principal Ideal Domain (PID)

Chapter 6: Gröbner Bases

Part II: Exercises and True or False? Questions

Chapter 7: Exercises

Chapter 8: True or False? Questions

Chapter 9: Solutions

Chapter 10: Further Reading

Part III: Appendix

Appendix A: Smith Canonical Form of Matrices

Appendix B: Gröbner Bases

Appendix C: Bibliography

Conclusion

This book is intended for advanced undergraduates or masters students, assuming only basic knowledge of finite fields, Abelian groups, and linear algebra. Its approach is constructive, aiming to illustrate concepts by applying them to the analysis of multivariate polynomial rings and modules over a principal ideal domain (PID) whenever feasible. Algorithms for computing these objects facilitate the generation of diverse examples. In particular, the structure of finitely generated modules over a PID is analyzed using the Smith canonical form of matrices. Furthermore, various properties of polynomial rings are investigated through the application of Buchbergers Algorithm for computing Gröbner bases. This book is intended to inspire the curiosity of readers and encourage them to find their own solutions to the exercises.

Weight: 756g
Dimension: 154 x 234 x 25 (mm)
ISBN-13: 9783031569098
Edition number: 2024 ed.