Intro
To the Reader
CONTENTS
INTRODUCTION
CHAPTER VIII. Semisimple Modules and Rings
1. ARTINIAN MODULES AND NOETHERIAN MODULES
1. Artinian Modules and Noetherian Modules
2. Artinian Rings and Noetherian Rings
3. Countermodule
4. Polynomials with Coefficients in a Noetherian Ring
Exercises
2. THE STRUCTURE OF MODULES OF FINITE LENGTH
1. Local Rings
2. Weyr-Fitting Decomposition
3. Indecomposable Modules and Primordial Modules
4. Semiprimordial Modules
5. The Structure of Modules of Finite Length
Exercises
3. SIMPLE MODULES

1. Simple Modules
2. Schur's Lemma
3. Maximal Submodules
4. Simple Modules over an Artinian Ring
5. Classes of Simple Modules
Exercises
4. SEMISIMPLE MODULES
1. Semisimple Modules
2. The homomorphism sum of homomorphisms
3. Some Operations on Modules
4. Isotypical Modules
5. Description of an Isotypical Module
6. Isotypical Components of a Module
7. Description of a Semisimple Module
8. Multiplicities and Lengths in Semisimple Modules
Exercises
5. COMMUTATION
1. The Commutant and Bicommutant of a Module
2. Generating Modules

3. The Bicommutant of a Generating Module
4. The Countermodule of a Semisimple Module
5. Density Theorem
6. Application to Field Theory
Exercises
6. MORITA EQUIVALENCE OF MODULES AND ALGEBRAS
1. Commutant and Duality
2. Generating Modules and Finitely Generated Projective Modules
3. Invertible Bimodules and Morita Equivalence
4. The Morita Correspondence of Modules
5. Ordered Sets of Submodules
6. Other Properties Preserved by the Morita Correspondence
7. Morita Equivalence of Algebras
Exercises
7. SIMPLE RINGS
1. Simple Rings

2. Modules over a Simple Ring
3. Degrees
4. Ideals of Simple Rings
Exercises
8. SEMISIMPLE RINGS
1. Semisimple Rings
2. Modules over a Semisimple Ring
3. Factors of a Semisimple Ring
4. Idempotents and Semisimple Rings
Exercises
9. RADICAL
1. The Radical of a Module
2. The Radical of a Ring
3. Nakayama's Lemma
4. Lifts of Idempotents
5. Projective Cover of a Module
Exercises
10. MODULES OVER AN ARTINIAN RING
1. The Radical of an Artinian Ring
2. Modules over an Artinian Ring
3. Projective Modules over an Artinian Ring
Exercises

11. GROTHENDIECK GROUPS Modules
1. Additive Functions of Modules
2. The Grothendieck Group of an Additive Set of Modules
3. Using Composition Series
4. The Grothendieck Group R(A)
5. Change of Rings
6. The Grothendieck Group R(A)
7. Multiplicative Structure on K(C)
8. The Grothendieck Group K(A)
9. The Grothendieck Group K(A) of an Artinian Ring
10. Change of Rings for K(A)
11. Frobenius Reciprocity
12. The Case of Simple Rings
Exercises
12. TENSOR PRODUCTS OF SEMISIMPLE MODULES
1. Semisimple Modules over Tensor Products of Algebras