A Course in Commutative Algebra

The author is professor of algorithmic algebra at the Technische Universität München, in Munich. He regularly teaches courses in commutative algebra, invariant theory, and computer algebra. In 2007 he received an award from the state of Bavaria for excellence in teaching.

Introduction.- Part I The Algebra Geometry Lexicon: 1 Hilbert's Nullstellensatz; 2 Noetherian and Artinian Rings; 3 The Zariski Topology; 4 A Summary of the Lexicon.- Part II Dimension: 5 Krull Dimension and Transcendence Degree; 6 Localization; 7 The Principal Ideal Theorem; 8 Integral Extensions.- Part III Computational Methods: 9 Gröbner Bases; 10 Fibers and Images of Morphisms Revisited; 11 Hilbert Series and Dimension.- Part IV Local Rings: 12 Dimension Theory; 13 Regular Local Rings; 14 Rings of Dimension One.- References.- Notation.- Index.

From the reviews:

"This recent addition to Springer's famous Graduate Texts in Mathematics (GTM) series comprises a thorough, modern introduction to commutative algebra with the central concepts and results almost exclusively motivated by their applications in algebraic geometry. ... The author writes in an engaging, reader-friendly manner. ... The main concepts are deftly presented and well-motivated and key ideas and methods are clearly highlighted. ... This textbook will be a very useful entrée for beginning graduate students ... ." (Nick Lord, The Mathematical Gazette, Vol. 97 (539), July, 2013)

"This book is an introductory text to commutative algebra with the idea also of being a guide to the algorithmic branch of the subject. ... this is a valuable and readable textbook on modern commutative algebra. It contains a huge number of exercises and it appeals to geometric intuition whenever possible. It can be highly recommended for independent reading or as material for preparation of courses." (Alejandro Melle Hernandez, The European Mathematical Society, July, 2012)

"This book gives a modern introduction to commutative algebra for students who had a first course in abstract algebra and who are familiar with the most basic notions of topology. ... The book can also serve as a first introduction to algebraic geometry. A strong feature of the book are the interesting exercises which nicely complement and illustrate the theory. Altogether this a stimulating book on a classical subject with an emphasis on the connection to algebraic geometry." (J. Mahnkopf, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)

"A student of algebraic geometry might naturally wish to make quick work of the algebraic prerequisites and progress. This volume's focused selection of material and geometrically informed viewpoint are tailored to this purpose. ... this is a useful, practical, and unintimidating book, especially suitable for advanced undergraduate students. Summing Up: Recommended. Upper-division undergraduates." ( D. V. Feldman, Choice, Vol. 48 (11), August, 2011)

"This is a well-written book that goes right away to the core of the subject: Commutative algebra as an introduction to algebraic geometry ... . This being said, this is a fine book, in the class of many other well established and fine books on the subject ... . I am sure it could become a text of choice for an introductory course in commutative algebra. Both the lecturer and the student would benefit from its balanced and novel approach to the subject." (Felipe Zaldivar, The Mathematical Association of America, March, 2011)

"This book provides a selection of basic material for an introductory graduate course in commutative algebra. ... it is intended to serve as a useful source for both instructors and students in the field, in particular also for the purpose of profound self-study. ... the exercises often come with directing hints helping the reader to tackle them successfully. ... Being really tailor-made for effective teaching and self-study, the text stands out by its expository mastery, didactic skill, versatility, expediency, and absolute reader-friendliness." (Werner Kleinert, Zentralblatt MATH, Vol. 1210, 2011)

"This book is a pleasant elementary course in commutative algebra with views toward algebraic geometry, computer algebra and invariant theory. ... It also studies Noetherian topological spaces and their irreducible components with an application to the spectrum of a Noetherian ring. ... Each chapter contains many exercises, and some of them are solved at the end of the book." (D.-M. Popescu, Mathematical Reviews, Issue 2011 j)