Graduate algebra: noncommutative view

Product Description

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. Gröbner 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.

Review

"The book is consistently organized in several layers (main text, supplements, appendices), which makes it a valuable source for readers of various levels, from graduate students to researchers." ---- European Mathematical Society Newsletter

"Its outstanding features are of many kinds, ranging from its underlying philosophy of keeping the material as much to the point as possible, thereby being highly efficient, up to the vast amount of topical extras that are barely found somewhere else. Already this first volume must be seen as a didactic masterpiece, profitable for both students and teachers likewise." ---- Zentralblatt MATH

"The author chose a very nice way to present all the basics needed to have a "solid basis in algebra", walking "direct to the goal" (my words), and showing the most important results of commutative algebra as soon and as simply as possible." ---- Mathematical Reviews