EXERCISES IN CLASSICAL RING THEORY

By Z. PANID

CONTENTS WEDDERBURN-ARTIN THEORY §1 BASIC TERMINOLOGY AND EXAMPLES 39 EXERCISES §2 SEMISIMPLICITY 13 EXERCISES §3 STRUCTURE OF SEMISIMPLE RINGS 27 EXERCISES JACOBSON RADICAL THEORY §4 THE JACOBSON RADICAL 38 EXERCISES §5 JACOBSON RADICAL UNDER CHANGE OF RINGS 15 EXERCISES §6 GROUP RINGS AND THE J-SEMISIMPLICITY PROBLEM 21 EXERCISES INTRODUCTION TO REPRESENTATION THEORY §7 MODULES OVER FINITE-DIMENSIONAL ALGEBRAS 10 EXERCISES §8 REPRESENTATIONS OF GROUPS 31 EXERCISES §9 LINEAR GROUPS 8 EXERCISES PRIME AND PRIMITIVE RINGS §10 THE PRIME RADICAL; PRIME AND SEMIPRIME RINGS 32 EXERCISES §11 STRUCTURE OF PRIMITIVE RINGS; THE DENSITY THEOREM 24 EXERCISES §12 SUBDIRECT PRODUCTS AND COMMUTATIVITY THEOREMS 32 EXERCISES INTRODUCTION TO DIVISION RINGS §13 DIVISION RINGS 19 EXERCISES §14 SOME CLASSICAL CONSTRUCTIONS 19 EXERCISES §15 TENSOR PRODUCTS AND MAXIMAL SUB?ELDS 6 EXERCISES §16 POLYNOMIALS OVER DIVISION RINGS 18 EXERCISES ORDERED STRUCTURES IN RINGS §17 ORDERINGS AND PREORDERINGS IN RINGS 15 EXERCISES §18 ORDERED DIVISION RINGS 7 EXERCISES LOCAL RINGS, SEMILOCAL RINGS, AND IDEMPOTENTS §19 LOCAL RINGS 17 EXERCISES §20 SEMILOCAL RINGS 20 EXERCISES §21 THE THEORY OF IDEMPOTENTS 35 EXERCISES §22 CENTRAL IDEMPOTENTS AND BLOCK DECOMPOSITIONS 13 EXERCISES PERFECT AND SEMIPERFECT RINGS §23 PERFECT AND SEMIPERFECT RINGS 11 EXERCISES §24 HOMOLOGICAL CHARACTERIZATIONS OF PERFECT AND SEMIPERFECT RINGS 10 EXERCISES §25 PRINCIPAL INDECOMPOSABLES AND BASIC RINGS