Steps In Commutative Algebra

This book meets the need for an introductory account of commutative algebra, and is aimed at advanced undergraduates and first-year graduate students with a background in basic abstract algebra. It provides a good foundation in commutative ring theory, from which the reader can proceed to more advanced works in commutative algebra and algebraic geometry. The style throughout is rigorous but concrete, with exercises and examples given within chapters at appropriate points and provided for the more challenging problems that are used in the subsequent development. After reminders about basic material on commutative rings, there is extensive discussion of ideals, and then of modules, with applications including canonical forms for square matrices. The development of the fundamental theory of commutative Noetherian rings is at the core of the book. Affine algebras over fields, dimension theory and regular local rings are also and for this second edition two further chapters, on regular sequences and Cohen-Macaulay rings, have been added. This book is ideal-for anyone who needs a grounding in commutative algebra.