(Mostly) Commutative Algebra

This book stems from lectures on commutative algebra for 4th-year university students at two French universities (Paris and Rennes). At that level, students have already followed a basic course in linear algebra and are essentially fluent with the language of vector spaces over fields. The topics introduced include arithmetic of rings, modules, especially principal ideal rings and the classification of modules over such rings, Galois theory, as well as an introduction to more advanced topics such as homological algebra, tensor products, and algebraic concepts involved in algebraic geometry. More than 300 exercises will allow the reader to deepen his understanding of the subject.

The book also includes 11 historical vignettes about mathematicians who contributed to commutative algebra.

Very well written text, with good, motivational examples and exercises, both somewhat theoretical as well as more practical, calculatory Includes many well-chosen exercises (of different degrees of difficulty!) that can easily be used as 'homework' Smooth and logical presentation by a specialist in the area with lots of teaching experience Perfect companion for both undergraduates and graduates

Autorentext
Antoine Chambert-Loir is a professor of mathematics at Université Paris Cité. His research addresses questions in algebraic geometry which are motivated by number theoretical problems. He is the author of two books published by Springer-Verlag: A Field Guide To Algebra, an introduction to Galois theory; and (Mostly) Commutative Algebra, an intermediate-level exposition of commutative algebra. With J. Nicaise and J. Sebag, he cowrote the research monograph Motivic Integration (published by Birkhäuser), which was awarded the 2017 Ferran Sunyer i Balaguer prize.

Inhalt

Preface.- 1 Rings.- 2 Ideals and divisibility.- 3 Modules.- 4 Field extensions.- 5 Modules over principal ideal rings.- 6 Noetherian and artinian rings. Primary decomposition.- 7 First steps in homological algebra.- 8 Tensor products and determinants.- 9 Commutative algebra: the normalization theorem, dimension theory, Dedekind rings.- Appendix.- References.- Index.-