Basic Notions of Algebra

From the reviews: "This is one of the few mathematical books, the reviewer has read from cover to cover ... The main merit is that nearly on every page you will find some unexpected insights..." (Zentralblatt fur Mathematik und Ihre Grenzgebiete, 1991) "...which I read like a novel and undoubtedly will become a classic. ... A merit of the book under review is that it contains several important articles from journals which are not all so easily accessible. ... Furthermore, at the end of the book, there are some Notes by the author which are indispensible for the necessary historical background information. ... This valuable book should be on the shelf of every algebraist and algebraic geometer." (Nieuw Archief voor Wiskunde, 1992) "... There are few proofs in full, but there is an exhilarating combination of sureness of foot and lightness of touch in the exposition ... which transports the reader effortlessly across the whole spectrum of algebra... The challenge to Ezekiel, "Can these bones live?" is, all too often, the reaction of students when introduced to the bare bones of the concepts and constructs of modern algebra. Shafarevich's book - which reads as comfortably as an extended essay - breathes life into the skeleton and will be of interest to many classes of readers..." (The Mathematical Gazette, 1991) "... According to the preface, the book is addressed to "students of mathematics in the first years of an undergraduate course, or theoretical physicists or mathematicians from outside algebra wanting to get an impression of the spirit of algebra and its place in mathematics." I think that this promise is fully justified. The beginner, the experts and also the interested scientist who had contact with algebraic notions - all will read this exceptional book with great pleasure and benefit." (Zeitschrift fur Kristallographie, 1991) "This is a truly wonderful book, one that anyone teaching algebra should read and which should be pointed out to talented students, particularly those who want to know a little more about what and why abstract algebra is. This book is volume 1 in the Algebra section of the Springer Encyclopedia of Mathematical Sciences ... . The examples are particularly well chosen, simple enough to understand... . one that will enrich your understanding of algebra and deepen your knowledge of mathematics as a whole." (Fernando Q. Gouvea, MathDL, March, 2007)

• What is Algebra?.- Fields.- Commutative Rings.- Homomorphisms and Ideals.- Modules.- Algebraic Aspects of Dimension.- The Algebraic View of Infinitesimal Notions.- Noncommutative Rings.- Modules over Noncommutative Rings.- Semisimple Modules and Rings.- Division Algebras of Finite Rank.- The Notion of a Group.- Examples of Groups: Finite Groups.- Examples of Groups: Infinite Discrete Groups.- Examples of Groups: Lie Groups and Algebraic Groups.- General Results of Group Theory.- Group Representations.- Some Applications of Groups.- Lie Algebras and Nonassociative Algebra.- Categories.- Homological Algebra.- K-theory.