The theory of rings is not only important in mathematics but in also in applications of algebra, such as cryptography, coding theory, and theoretical physics. The theory of rings has advanced considerably since this book was published, but the book emphasizes the fundamentals, which have not changed, and so could still be used as an introduction to these modern developments. The book is written for undergraduates who have had prior exposure to modern algebra.There are many exercises at the end of each chapter, and hints of some of their solutions in the back of the book. Ring theory used to be viewed as an abstract, esoteric subject, but this attitude has changed due to its many applications. Commutative rings are the emphasis in the book, and this reflects somewhat its time of publication. Noncommutative rings have been the subject of intense investigation in the last decade, going by the name of quantum algebra. Quantum algebra has its origins in theory of exactly solved models in statistical mechanics, but has now established itself as a rigorous subfield of modern algebra. Readers interested in this area of research will be well prepared after a study of this book. Some of the highlights of the book, a few of which are usually not included in a beginning course in ring theory include the discussions on ideals in complete metric rings, dense rings, and modular ideals.