In this new edition of "First Course", the entire text has been retyped,some proofs were rewritten, and numerous improvements in the exposition have been included. The original chapters and sections have remained unchanged, with the exception of the addition of an Appendix (on uniserial modules) to 20. All known typographical errors were corrected (although no doubt a few new ones have been introduced in the process!). The original exercises in the first edition have been replaced by the 400 exercises in the problem book (Lam [95]) , and I have added at least 30 more in this edition for the convenience of the reader. As before, the book should be suitable as a text for a one-semester or a full-year graduate course in noncommutative ring theory.

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, primitive and semiprimitive rings, division rings, ordered rings, local and semilocal rings, perfect and semiperfect rings, and so forth. 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.

preface to the second edition

preface to the first edition

notes to the reader

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-semisimplicjty 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, semilocai rings, and idempotents

19. local rings

exercises for 19

20. semilocal rings

appendix: endomorphism rings of uniserial modules

exercises for 20

21. th theory ofidempotents

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. homoiogical characterizations of perfect and semiperfect rings

exercises for 24

25. principal indecomposables and basic rings

exercises for 25

references

name index

subject index