a First Course in Rings And Ideals

This volume is designed to serve as an introduction to the basic ideas and techniques of ring theory. It is intended to be an expository textbook, rather than a treatise on the subject. The mathematical background required for a proper understanding of the contents is not extensive. We assume that the average reader has had some prior contact with abstract algebra but is still relatively inexperienced in this respect.

This text introduces readers to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each.

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).

This book presents a graduate-level course on modern algebra. It can be used as a teaching book – owing to the copious exercises – and as a source book for those who wish to use the major theorems of algebra.

This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level.