This volume studies commutative rings, ideals, modules, and their structural properties.
This volume studies commutative rings, ideals, modules, and their structural properties. It forms the algebraic foundation for algebraic geometry and number theory.
Part I. Rings and Ideals
Chapter 1. Commutative Rings
1.1 Definitions and examples 1.2 Ring homomorphisms 1.3 Subrings and quotient rings 1.4 Units and zero divisors 1.5 Basic constructions
Chapter 2. Ideals
2.1 Definition and examples 2.2 Prime and maximal ideals 2.3 Operations on ideals 2.4 Ideal quotient and radical 2.5 Correspondence theorems
Chapter 3. Ring Constructions
3.1 Polynomial rings 3.2 Localization 3.3 Product rings 3.4 Extension and contraction 3.5 Examples
Part II. Noetherian Rings
Chapter 4. Chain Conditions
4.1 Ascending chain condition 4.2 Noetherian rings 4.3 Finitely generated ideals 4.4 Hilbert basis theorem 4.5 Applications
Chapter 5. Modules over Rings
5.1 Definitions and examples 5.2 Submodules and quotient modules 5.3 Homomorphisms 5.4 Exact sequences 5.5 Free and projective modules
Chapter 6. Primary Decomposition
6.1 Primary ideals 6.2 Decomposition theorems 6.3 Associated primes 6.4 Uniqueness properties 6.5 Applications
Part III. Dimension Theory
Chapter 7. Krull Dimension
7.1 Chains of prime ideals 7.2 Definition of dimension 7.3 Examples 7.4 Dimension of polynomial rings 7.5 Applications
Chapter 8. Integral Extensions
8.1 Integral elements 8.2 Integral closure 8.3 Going-up and going-down 8.4 Normal domains 8.5 Applications
Chapter 9. Valuation Theory
9.1 Valuation rings 9.2 Discrete valuations 9.3 Valuation ideals 9.4 Completions 9.5 Applications
Part IV. Local Algebra
Chapter 10. Local Rings
10.1 Definition and examples 10.2 Maximal ideals 10.3 Localization revisited 10.4 Residue fields 10.5 Applications
Chapter 11. Nakayama’s Lemma
11.1 Statement and proof 11.2 Consequences 11.3 Applications to modules 11.4 Generators and relations 11.5 Structural insights
Chapter 12. Completions
12.1 Adic topology 12.2 Completion of rings 12.3 Formal power series 12.4 Applications 12.5 Examples
Part V. Homological Methods
Chapter 13. Exact Sequences
13.1 Short and long exact sequences 13.2 Diagram chasing 13.3 Snake lemma 13.4 Five lemma 13.5 Applications
Chapter 14. Derived Functors
14.1 Tor functor 14.2 Ext functor 14.3 Projective and injective resolutions 14.4 Homological dimension 14.5 Applications
Chapter 15. Depth and Regularity
15.1 Regular sequences 15.2 Depth of modules 15.3 Cohen–Macaulay rings 15.4 Regular local rings 15.5 Applications
Part VI. Advanced Ring Theory
Chapter 16. Integral Domains
16.1 Unique factorization domains 16.2 Principal ideal domains 16.3 Dedekind domains 16.4 Class groups 16.5 Applications
Chapter 17. Completion and Formal Methods
17.1 Formal schemes (overview) 17.2 Adic rings 17.3 Flatness 17.4 Completion techniques 17.5 Applications
Chapter 18. Flatness and Tensor Products
18.1 Tensor products 18.2 Flat modules 18.3 Faithful flatness 18.4 Base change 18.5 Applications
Part VII. Connections and Applications
Chapter 19. Algebraic Geometry Interface
19.1 Coordinate rings 19.2 Spec of a ring 19.3 Zariski topology 19.4 Sheaf perspective (overview) 19.5 Applications
Chapter 20. Number Theory Interface
20.1 Rings of integers 20.2 Localization in number theory 20.3 Valuations and primes 20.4 Ideal factorization 20.5 Applications
Chapter 21. Computational Commutative Algebra
21.1 Gröbner bases 21.2 Buchberger algorithm 21.3 Polynomial ideals 21.4 Computational complexity 21.5 Software tools
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Tight closure (overview) 22.2 Homological conjectures 22.3 Singularities 22.4 Derived algebraic geometry (overview) 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Dimension theory questions 23.2 Homological bounds 23.3 Classification problems 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of commutative algebra 24.2 Key contributors 24.3 Evolution of methods 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common identities and lemmas B. Standard examples reference C. Proof techniques checklist D. Algorithm templates E. Cross-reference to other MSC branches
This volume provides the algebraic core used in geometry and number theory. It emphasizes structure, local behavior, and homological methods.