Here are 100 chapter titles on Ring Theory, progressing from beginner to advanced levels, emphasizing the mathematical aspects:
I. Foundations and Basic Concepts (1-20)
- Introduction to Rings: Definitions and Examples
- Rings with Unity and Rings without Unity
- Commutative Rings and Non-Commutative Rings
- Zero Divisors and Integral Domains
- Fields: A Special Kind of Ring
- Subrings and Ideals: Basic Definitions
- Examples of Rings: Integers, Polynomials, Matrices
- Ring Homomorphisms: Preserving the Structure
- Isomorphisms of Rings: When Rings are the Same
- Kernel and Image of a Ring Homomorphism
- Quotient Rings: Building New Rings from Old
- The Isomorphism Theorems for Rings
- Direct Products of Rings
- Characteristic of a Ring
- Prime Ideals and Maximal Ideals
- Nilpotent Elements and their Properties
- Units in a Ring: The Invertible Elements
- Division Rings and Skew Fields
- Polynomial Rings: Formal Power Series
- Practice Problems: Basic Ring Theory
II. Special Rings and Ideals (21-40)
- Principal Ideal Domains (PIDs)
- Euclidean Domains: Division with Remainder
- Unique Factorization Domains (UFDs)
- The Relationship Between EDs, PIDs, and UFDs
- Polynomial Rings over Fields: A Deeper Look
- Irreducible Polynomials and Factorization
- Gauss's Lemma and its Applications
- Noetherian Rings and Artinian Rings
- Hilbert Basis Theorem: Polynomial Rings are Noetherian
- Chain Conditions on Ideals
- Prime and Maximal Ideals in Noetherian Rings
- Radical of an Ideal and Nilradical of a Ring
- Jacobson Radical and its Properties
- Local Rings: Rings with a Unique Maximal Ideal
- Chinese Remainder Theorem: Generalizations
- Applications: CRT in Number Theory and Cryptography
- Fractional Ideals and Dedekind Domains
- Dedekind Domains and Unique Factorization of Ideals
- Valuation Rings: Introduction and Basic Properties
- Practice Problems: Special Rings and Ideals
III. Modules and Representations (41-60)
- Introduction to Modules: Definitions and Examples
- Module Homomorphisms and Isomorphisms
- Submodules and Quotient Modules
- Direct Sums and Direct Products of Modules
- Free Modules: Modules with a Basis
- Finitely Generated Modules
- Torsion Modules and Torsion-Free Modules
- Modules over a PID: Structure Theorem
- Exact Sequences of Modules
- Projective Modules and Injective Modules
- Flat Modules: Introduction and Properties
- Tensor Product of Modules: Definition and Properties
- Algebras: Rings that are also Vector Spaces
- Representations of Groups: Introduction
- Group Rings and their Properties
- Representations of Algebras
- Simple Modules and Semisimple Modules
- Schur's Lemma: A Fundamental Result
- Characters of Representations
- Practice Problems: Modules and Representations
IV. Field Theory and Galois Theory (61-80)
- Field Extensions: Basic Concepts
- Algebraic Extensions and Transcendental Extensions
- Splitting Fields and Normal Extensions
- Separable Extensions and Inseparable Extensions
- The Fundamental Theorem of Galois Theory
- Galois Groups: Groups of Automorphisms
- Cyclotomic Fields and their Galois Groups
- Solvable Groups and Solvability by Radicals
- Impossibility of Angle Trisection and Squaring the Circle
- Constructible Numbers: A Field-Theoretic Approach
- Finite Fields: Structure and Properties
- Applications: Finite Fields in Coding Theory
- Algebraic Closure and its Existence
- Transcendental Extensions: A Deeper Look
- Infinite Galois Extensions
- Profinite Groups and their Properties
- Valuation Theory: Extensions of Valuations
- Dedekind Domains and Valuations
- Arithmetic of Number Fields: Ideal Class Group
- Practice Problems: Field and Galois Theory
V. Advanced Topics and Applications (81-100)
- Non-Commutative Ring Theory: Introduction
- Wedderburn's Theorem: Structure of Semisimple Rings
- Artinian Rings and their Properties
- Hopkins-Levitzki Theorem
- Division Algebras and their Classification
- K-Theory: Introduction and Basic Concepts
- Algebraic Geometry and Ring Theory: Connections
- Commutative Algebra: Advanced Topics
- Homological Algebra: Introduction and Basic Concepts
- Category Theory: Basic Definitions and Examples
- Functors and Natural Transformations
- Applications: Ring Theory in Cryptography
- Applications: Ring Theory in Coding Theory
- Applications: Ring Theory in Physics
- Ring Theory and Representation Theory: Advanced Topics
- Quantum Groups: Introduction and Basic Concepts
- Hopf Algebras: Introduction and Examples
- Non-commutative Geometry: Introduction
- Research Trends in Ring Theory
- The Future of Ring Theory and its Applications