A First Course in Noncommutative Rings (Interdisciplinary Applied Mathematics)

One of my favorite graduate courses at Berkeley is Math 251, a one-semester course in ring theory offered to second-year level graduate students. I taught this course in the Fall of 1983, and more recently in the Spring of 1990, both times focusing on the theory of noncommutative rings. This book is an outgrowth of my lectures in these two courses, and is intended for use by instructors and graduate students in a similar one-semester course in basic ring theory. Ring theory is a subject of central importance in algebra. Historically, some of the major discoveries in ring theory have helped shape the course of development of modern abstract algebra. Today, ring theory is a fer­ tile meeting ground for group theory (group rings), representation theory (modules), functional analysis (operator algebras), Lie theory (enveloping algebras), algebraic geometry (finitely generated algebras, differential op­ erators, invariant theory), arithmetic (orders, Brauer groups), universal algebra (varieties of rings), and homological algebra (cohomology of rings, projective modules, Grothendieck and higher K-groups). In view of these basic connections between ring theory and other branches of mathemat­ ics, it is perhaps no exaggeration to say that a course in ring theory is an indispensable part of the education for any fledgling algebraist. The purpose of my lectures was to give a general introduction to the theory of rings, building on what the students have learned from a stan­ dard first-year graduate course in abstract algebra. A First Course In Noncommutative Rings, An Outgrowth Of The Author's Lectures At The University Of California At Berkeley, Is Intended As A Textbook For A One-semester Course In Basic Ring Theory. By Aiming The Level Of Writing At The Novice Rather Than The Connoisseur And By Stressing The Role Of Examples And Motivation, The Author Has Produced A Text That Is Suitable Not Only For Use In A Graduate Course, But Also For Self-study In The Subject By Interested Graduate Students. More Than 400 Exercises Testing The Understanding Of The General Theory In The Text Are Included In This New Edition.--jacket. Machine Generated Contents Note: Chapter 1 -- Wedderburn-artin Theory --1. Basic Terminology And Examples -- Exercises For 1 -- 2. Semisimplicity -- Exercises For 2 -- 3. Structure Of Semisimple Rings -- Exercises For 3 --chapter 2 -- Jacobson Radical Theory --4. The Jacobson Radical -- Exercises For 4 -- 5. Jacobson Radical Under Change Of Rings -- Exercises For 5 -- 6. Group Rings And The J-semisimplicity Problem -- Exercises For 6 --chapter 3 -- Introduction To Representation Theory --7. Modules Over Finite-dimensional Algebras -- Exercises For 7 --8. Representations Of Groups -- Exercises For 8 -- 9. Linear Groups -- Exercises For 9 Chapter 4 -- Prime And Primitive Rings --10. The Prime Radical; Prime And Semiprime Rings -- Exercises For 10 -- 11. Structure Of Primitive Rings; The Density Theorem -- Exercises For 11 -- 12. Subdirect Products And Commutativity Theorems -- Exercises For 12 Chapter 5 -- Introduction To Division Rings --13. Division Rings -- Exercises For 13 -- 14. Some Classical Constructions -- Exercises For 14 -- 15. Tensor Products And Maximal Subfields -- Exercises For 15 -- 16. Polynomials Over Division Rings -- Exercises For 16 Chapter 6 -- Ordered Structures In Rings --17. Orderings And Preorderings In Rings -- Exercises For 17 -- 18. Ordered Division Rings -- Exercises For 18 Chapter 7 -- Local Rings, Semilocal Rings, And Idempotents --19. Local Rings -- Exercises For 19 -- 20. Semilocal Rings -- Appendix: Endomorphism Rings Of Uniserial Modules -- Exercises For 20 -- 21. Th Theory Of Idempotents -- Exercises For 21 -- 22. Central Idempotents And Block Decompositions -- Exercises For 22 -- Chapter 8 -- Perfect And Semiperfect Rings --23. Perfect And Semiperfect Rings -- Exercises For 23 -- 24. Homological Characterizations Of Perfect And Semiperfect Rings -- Exercises For 24 -- 25. Principal Indecomposables And Basic Rings -- Exercises For 25. T.y. Lam. Includes Bibliographical References And Indexes. A First Course in Noncommutative Rings, an outgrowth of the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-semester course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semisimple rings, Jacobson's theory of the radical, representation theory of groups and algebras, prime and semiprime rings, local and semilocal rings, perfect and semiperfect rings, etc. By aiming the level of writing at the novice rather than the connoisseur and by stressing th the role of examples and motivation, the author has produced a text that is suitable not only for use in a graduate course, but also for self- study in the subject by interested graduate students. More than 400 exercises testing the understanding of the general theory in the text are included in this new edition. Front Matter....Pages i-xv Wedderburn-Artin Theory....Pages 1-50 Jacobson Radical Theory....Pages 51-105 Introduction to Representation Theory....Pages 107-162 Prime and Primitive Rings....Pages 163-212 Introduction to Division Rings....Pages 213-274 Ordered Structures in Rings....Pages 275-292 Local Rings, Semilocal Rings, and Idempotents....Pages 293-343 Perfect and Semiperfect Rings....Pages 345-380 Back Matter....Pages 381-400 This textbook, drawn from the author's lectures at the University of California, is intended for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings and Jacobson's theory of radical representation of groups and algebras.