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This book is an introduction to the theory of noncommutative
algebra. The core of the book is suitable for a one-semester course
for graduate students. The approach, which is more homological than
ring-theoretic, clarifies the subject and its relation to other
important areas of mathematics, including K-theory, homological
algebra, and representation theory. The main part of the book
begins with a brief review of background material; the first
chapter covers the basics of semisimple modules and rings,
including the Wedderburn structure theorem; chapter two discusses
the Jacobson radical, giving several different views; chapter three
develops the theory of central simple algebras, including proofs of
the Skolem-Noether and Double Centralizer theorems, with two famous
theorems of Wedderburn and Frobenius given as applications; and
chapter four is an introduction to the Brauer group and its
relation to cohomology. The remaining chapters introduce several
special topics: the notion of primitive ring 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. Problems throughout the
book provide concrete examples, applications and amplifications of
the text; a set of supplementary problems explores further topics
and can serve as starting points for student projects.
About This Book This book is meant to be used by beginning graduate
students. It covers basic material needed by any student of
algebra, and is essential to those specializing in ring theory,
homological algebra, representation theory and K-theory, among
others. It will also be of interest to students of algebraic
topology, functional analysis, differential geometry and number
theory. Our approach is more homological than ring-theoretic, as
this leads the to many important areas of mathematics. This ap
student more quickly proach is also, we believe, cleaner and easier
to understand. However, the more classical, ring-theoretic
approach, as well as modern extensions, are also presented via
several exercises and sections in Chapter Five. We have tried not
to leave any gaps on the paths to proving the main theorem- at most
we ask the reader to fill in details for some of the sideline
results; indeed this can be a fruitful way of solidifying one's
understanding."
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