Noncommutative Rings

Description

A classic advanced textbook, containing a cross-section of ideas, techniques and results that give the reader an unparalleled introductory overview of the subject. The author gives an integrated presentation of overall theory and its applications in, for example, the study of groups of matrices, group representations, and in settling the problems of Burnside and Kurosh. Readers are also informed of open questions. Definitions are kept to a minimum and the statements of the theorems are sharp and clear. PREFACE CONTENTS 1. THE JACOBSON RADICAL 1. Modules 2. The radical of a ring 3. Artinian rings 4. Semisimple Artinian rings References 2. SEMISIMPLE RINGS 1. The density theorem 2. Semisimple rings 3. Applications of Wedderburn's theorem References 3. COMMUTATIVITY THEOREMS 1. Wedderburn's Theorem and some generalizations 2. Some special rings References 4. SIMPLE ALGEBRAS 1. The Brauer group 2. Maximal subfields 3. Some classic theorems 4. Crossed products References 5. REPRESENTATIONS OF FINITE GROUPS 1. The elements of the theory 2. A theorem of Hurwitz 3. Applications to group theory References 6. POLYNOMIAL IDENTITIES 1. A result on radicals 2. Standard identities 3. A theorem of Kaplansky 4. The Kurosh Problem for P.I. algebras References 7. GOLDIE'S THEOREM 1. Ore's theorem 2. Goldie's theorems 3. Ultra-products and a theorem of Posner References 8. THE GOLOD-SHAFAREVITCH THEOREM References SUBJECT INDEX NAME INDEX AFTERWORD by Lance Small