Elements of Mathematics

This book is a reprint of the English translation of Bourbaki's commutative algebra, chapters 1-7. It gives a rather complete treatment of commutative algebra, enabling one to go further and study algebraic or arithmetical geometry. The first three chapters successively treat the notions of flatness, localization and completions (in the general setting of graduations and filtrations). Chapters IV studies associated prime ideals and the primary decomposition. Chapter V deals with integers, integral closures and finitely generated algebras over a field (including the Nullstellensatz). Chapter VI studies valuation (of any rank), and the last chapter centers on divisors (Krull, Dedekind or factorial domains), with a final section on modules over integrally closed Noetherian domains, not usually found in textbooks. A historical note is appended, and useful exercises are gathered at the end of chapters. Four talks recall the main properties of commutative rings established in the text.