Advanced Algebra

This book's organizing principle is the interplay between groups and rings, where "rings" includes the ideas of modules. It contains basic denitions, complete and clear theorems (the rst with brief sketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of denition-theorem-proofs, this text put results and ideas in context so that students can appreciate why a certain topic is being studied, and where denitions originate. Advanced Algebra systematically develops concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. The book aims to give the reader a global view of algebra, its use, and its role in mathematics as a whole. The idea is to explain what the young mathematician needs to know about algebra in order to communicate well with colleagues in all branches of mathematics. The book is written as textbooks, and the primary audience is students who are learning the material for the rst time and who are planning a career in which they will use advanced mathematics professionally. Much of the material in the book, including nearly all of Basic Algebra and some of Advanced Algebra, corresponds to normal course work, with the proportions depending on the university. The books include further topics that may be skipped in required courses but that the professional mathematician will ultimately want to learn by self-study. This book can be used as a text for the rst year of graduate algebra, but it is much more than that. It can also serve more advanced graduate students wishing to learn topics on their own; while not reaching the frontiers, the book does provide a sense of the successes and methods arising in an area. Finally, this is a reference containing many of the standard theorems and denitions that users of algebra need to know. Thus, the book is not only an appetizer, but a hearty meal as well.