This volume studies rings and algebras with associative multiplication, without requiring commutativity.
This volume studies rings and algebras with associative multiplication, without requiring commutativity. It develops structure theory, module theory, and connections to representation theory and geometry.
Part I. Basic Ring Theory
Chapter 1. Rings and Homomorphisms
1.1 Definitions and examples 1.2 Subrings and ideals 1.3 Ring homomorphisms 1.4 Quotient rings 1.5 Basic constructions
Chapter 2. Ideals and Structure
2.1 Left, right, and two-sided ideals 2.2 Prime and maximal ideals 2.3 Jacobson radical 2.4 Nilpotent elements 2.5 Examples
Chapter 3. Modules over Rings
3.1 Definitions and examples 3.2 Submodules and quotient modules 3.3 Homomorphisms 3.4 Direct sums 3.5 Exact sequences
Part II. Structure Theory
Chapter 4. Artinian and Noetherian Rings
4.1 Chain conditions 4.2 Artinian rings 4.3 Noetherian rings 4.4 Hopkins–Levitzki theorem 4.5 Applications
Chapter 5. Semisimple Rings
5.1 Simple modules 5.2 Semisimplicity 5.3 Wedderburn’s theorem 5.4 Matrix rings 5.5 Structure decomposition
Chapter 6. Radicals
6.1 Jacobson radical 6.2 Nilradical 6.3 Prime radical 6.4 Radical properties 6.5 Applications
Part III. Representation Theory
Chapter 7. Modules and Representations
7.1 Representations of algebras 7.2 Irreducible modules 7.3 Decomposition 7.4 Endomorphism rings 7.5 Examples
Chapter 8. Algebras over Fields
8.1 Finite-dimensional algebras 8.2 Structure constants 8.3 Basic algebras 8.4 Path algebras (overview) 8.5 Applications
Chapter 9. Representation Types
9.1 Finite representation type 9.2 Tame and wild algebras 9.3 Classification problems 9.4 Examples 9.5 Applications
Part IV. Polynomial and Matrix Rings
Chapter 10. Polynomial Rings over Rings
10.1 Definitions 10.2 Properties 10.3 Factorization issues 10.4 Extensions 10.5 Examples
Chapter 11. Matrix Rings
11.1 Full matrix algebras 11.2 Properties 11.3 Similarity 11.4 Canonical forms (overview) 11.5 Applications
Chapter 12. Division Rings
12.1 Definitions 12.2 Examples 12.3 Skew fields 12.4 Central simple algebras (overview) 12.5 Applications
Part V. Homological Methods
Chapter 13. Projective and Injective Modules
13.1 Definitions 13.2 Properties 13.3 Resolutions 13.4 Applications 13.5 Examples
Chapter 14. Derived Functors
14.1 Ext and Tor 14.2 Homological dimension 14.3 Global dimension 14.4 Applications 14.5 Examples
Chapter 15. Homological Algebra in Rings
15.1 Exact sequences revisited 15.2 Derived categories (overview) 15.3 Triangulated structures 15.4 Applications 15.5 Connections
Part VI. Noncommutative Structures
Chapter 16. Noncommutative Polynomial Rings
16.1 Free algebras 16.2 Relations and presentations 16.3 Gröbner bases (noncommutative) 16.4 Applications 16.5 Examples
Chapter 17. Ore Extensions
17.1 Skew polynomial rings 17.2 Differential operators 17.3 Quantum analogues 17.4 Applications 17.5 Examples
Chapter 18. Quantum Algebras (Overview)
18.1 Basic concepts 18.2 Deformations 18.3 Quantum groups 18.4 Representations 18.5 Applications
Part VII. Connections and Applications
Chapter 19. Ring Theory in Geometry
19.1 Noncommutative geometry (overview) 19.2 Coordinate algebras 19.3 Deformations 19.4 Applications 19.5 Examples
Chapter 20. Ring Theory in Physics
20.1 Operator algebras 20.2 Quantum mechanics 20.3 Symmetry algebras 20.4 Applications 20.5 Examples
Chapter 21. Computational Aspects
21.1 Algorithmic ring theory 21.2 Computation with modules 21.3 Symbolic algebra systems 21.4 Complexity considerations 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Noncommutative algebraic geometry 22.2 Higher representation theory 22.3 Derived algebraic structures 22.4 Categorification (overview) 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Classification of algebras 23.2 Representation complexity 23.3 Homological conjectures 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of ring theory 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common identities and constructions B. Standard examples reference C. Proof techniques checklist D. Algorithm templates E. Cross-reference to other MSC branches
This volume builds the theory of associative rings and algebras as a central noncommutative framework. It connects algebra, representation theory, and applications in physics and computation.