Constantin Nastasescu

Methods of graded rings

The Category of Graded Rings.- The Category of Graded Modules.- Modules over Stronly Graded Rings.- Graded Clifford Theory.- Internal Homogenization.- External Homogenization.- Smash Products.- Localization of Graded Rings.- Application to Gradability.- Appendix A: Some Category Theory.- Appendix B: Dimensions in an Abelian Category.- Bibliography.- Index.

9. Application to Gradability

9.1 General Descent Theory 9.2 Good gradings on matrix algebras 9.3 Gradings over cyclic groups 9.4 C2-gradings of Open image in new window 9.5 C2-gradings on Open image in new window in characteristic 2 9.6 Gradability of modules 9.7 Comments and References for Chapter 9.

8. Localization of Graded Rings

8.1 Graded rings of fractions 8.2 Localization of Graded Rings for a Graded Linear Topology 8.3 Graded Rings and Modules of Quotients 8.4 The Graded Version of Goldie’s Theorem 8.5 Exercises 8.6 Comments and References for Chapter 8

7. Smash Products

7.1 The Construction of the Smash Product 7.2 The Smash Product and the Ring Open image in new window 7.3 Some Functorial Constructions 7.4 Smash Product and Finiteness Conditions 7.5 Prime Ideals of Smash Products 7.6 Exercises 7.7 Comments and References for Chapter 7

6. External Homogenization

6.1 Normal subsemigroup of a group 6.2 External homogenization 6.3 A Graded Version of Maschke’s Theorem. Applications 6.4 Homogenization and Dehomogenization Functors 6.5 Exercises 6.6 Comments and References for Chapter 6

5. Internal Homogenization

5.1 Ordered Groups 5.2 Gradation by Ordered Groups. Elementary Properties 5.3 Internal Homogenization 5.4 Chain Conditions for Graded Modules 5.5 Krull Dimension of Graded Rings 5.6 Exercises 5.7 Comments and References for Chapter 5

4. Graded Clifford Theory

4.1 The Category Mod Open image in new window 4.2 The Structure of Objects of Mod Open image in new window as R e -modules 4.3 The Classical Clifford Theory for Strongly Graded Rings 4.4 Application to Graded Clifford Theory 4.5 Torsion Theory and Graded Clifford Theory 4.6 The Density Theorem for gr-simple modules 4.7 Extending (Simple) Modules 4.8 Exercises 4.9 Comments and References for Chapter 4

3. Modules over Strongly Graded Rings

3.1 Dade’s Theorem 3.2 Graded Rings with R-gr Equivalent to R e -mod 3.3 Strongly Graded Rings over a Local Ring 3.4 Endomorphism G-Rings 3.5 The Maschke Theorem for Strongly Graded Rings 3.6 H-regular Modules 3.7 Green Theory for Strongly Graded Rings 3.8 Exercises 3.9 Comments and References for Chapter 3

2. The Category of Graded Modules

2.1 Graded Modules 2.2 The category of Graded Modules 2.3 Elementary Properties of the Category R-gr 2.4 The functor Open image in new window 2.5 Some Functorial Constructions 2.6 Some Topics in Torsion Theory on R-gr 2.7 The Structure of Simple Objects in R-gr 2.8 The Structure of Gr-injective Modules 2.9 The Graded Jacobson Radical (Graded Version of Hopkins’ Theorem) 2.10 Graded Endomorphism Rings and Graded Matrix Rings 2.11 Graded Prime Ideals. The Graded Spectrum 2.12 Exercises 2.13 Comments and References for Chapter 2

1. The Category of Graded Rings

1.1 Graded Rings 1.2 The Category of Graded Rings 1.3 Examples 1.4 Crossed Products 1.5 Exercises 1.6 Comments and References for Chapter 1