This book is an expanded text for a graduate course in commutative algebra, focusing on the algebraic underpinnings of algebraic geometry and of number theory. Accordingly, the theory of affine algebras is featured, treated both directly and via the theory of Noetherian and Artinian modules, and the theory of graded algebras is included to provide the foundation for projective varieties. Major topics include the theory of modules over a principal ideal domain, and its applications to matrix theory (including the Jordan decomposition), the Galois theory of field extensions, transcendence degree, the prime spectrum of an algebra, localization, and the classical theory of Noetherian and Artinian rings.Later chapters include some algebraic theory of elliptic curves (featuring the Mordell-Weil theorem) and valuation theory, including local fields. One feature of the book is an extension of the text through a series of appendices. This permits the inclusion of more advanced material, such as transcendental field extensions, the discriminant and resultant, the theory of Dedekind domains, and basic theorems of rings of algebraic integers. An extended appendix on derivations includes the Jacobian conjecture and Makar-Limanov's theory of locally nilpotent derivations. Grobner bases can be found in another appendix. Exercises provide a further extension of the text. The book can be used both as a textbook and as a reference source.
Introduction to modules and their structure theory Introduction and prerequisites Exercises-Chapter 0 Part I. Modules: Finitely generated modules Simple modules and composition series Exercises-Part I Part II. Affine algebras and Noetherian rings: Galois theory of fields Algebras and affine fields Transcendence degree and the Krull dimension of a ring Modules and rings satisfying chain conditions Localization and the prime spectrum The Krull dimension theory of commutative Noetherian rings Exercises-Part II Part III. Applications to geometry and number theory: The algebraic foundations of geometry Applications to algebraic geometry over the rationals -- Diophantine equations and elliptic curves Absolute values and valuation rings Exercises-Part III List of major results Bibliography Index.