A Course in Commutative Algebra [Hardcover]

This textbook offers a thorough, modern introduction into commutative algebra. It is intented mainly to serve as a guide for a course of one or two semesters, or for self-study. The carefully selected subject matter concentrates on the concepts and results at the center of the field. The book maintains a constant view on the natural geometric context, enabling the reader to gain a deeper understanding of the material. Although it emphasizes theory, three chapters are devoted to computational aspects. Many illustrative examples and exercises enrich the text.This text offers a thorough, modern introduction to commutative algebra. It concentrates on concepts and results at the center of the field while keeping a constant view on the natural geometrical context. It includes many examples and exercises.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 Gr?bner 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 Springers 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 entr?e for beginning graduate students & . (Nick Lord, The Mathematical Gazette, Vol. 97 (539), July, 2013)

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