Graduate Texts in Mathematics

This book provides a concise introduction to the field of noncommutative algebra. It covers material essential to all students of algebra, particularly those specializing in ring theory, homological algebra, representation theory and K-theory. The core of the book is suitable for a one-semester graduate course; it is also suitable for self-study.
The approach is more homological than ring-theoretic and begins with the basics of semisimple modules and rings, including the Wedderburn structure theorem. The Jacobson radical is discussed from several different points of view followed by a development of the theory of central simple algebras, including proofs of the Skolem-Noether and Double Centralizer theorems. This theory is then used to give quick proofs of two famous, classical results: the Wedderburn Theorem on finite division rings and the Frobenius Theorem on the classification of central division algebras over the reals. The first part of the book closes with an introduction to the Brauer group and its relation to cohomology.
The remaining chapters consist of special topics: the notion of primitive rings is developed along lines parallel to that of simple rings; the representation theory of finite groups is combined with the Wedderburn Structure Theorem to prove Burnside's Theorem; the global dimension of a ring is studied using Kaplansky's elementary point of view; and the Brauer group of a commutative ring is introduced. In addition to the large number of exercises throughout the book, a set of supplementary problems explores further topics and can serve as a starting point for student projects.