Book
This book is an introduction to the theory of noncommutative algebra. The coreof the book is suitable for a onesemester course for graduate students. Theapproach which is more homological than ringtheoretic clarifies the subjectand its relation to other important areas of mathematics including Ktheoryhomological algebra and representation theory. The main part of the bookbegins with a brief review of background material the first chapter covers thebasics of semisimple modules and rings including the Wedderburn structuretheorem chapter two discusses the Jacobson radical giving several differentviews chapter three develops the theory of central simple algebras includingproofs of the SkolemNoether and Double Centralizer theorems with two famoustheorems of Wedderburn and Frobenius given as applications and chapter four isan introduction to the Brauer group and its relation to cohomology. Theremaining chapters introduce several special topics the notion of primitivering is developed along lines parallel to that of simple rings therepresentation theory of finite groups is combined with the WedderburnStructure Theorem to prove Burnsides Theorem the global dimension of a ringis studied using Kaplanskys elementary point of view and the Brauer group ofa commutative ring is introduced. Problems throughout the book provide concreteexamples applications and amplifications of the text a set of supplementaryproblems explores further topics and can serve as starting points for studentprojects. «
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