Ring Theory
A 24-module path through ring theory, in study order from foundations to the frontier.
Modules
- Rings: Definition and Examples
- Commutative Rings, Identity, and Units
- Zero Divisors and Integral Domains
- Fields
- Characteristic of a Ring
- Subrings
- Ideals
- Principal Ideals and Ideal Generation
- Quotient Rings
- Ring Homomorphisms
- The Isomorphism Theorems for Rings
- Operations on Ideals: Sum, Product, Intersection
- Prime Ideals
- Maximal Ideals and Fields
- The Field of Fractions
- The Chinese Remainder Theorem
- Polynomial Rings
- The Division Algorithm for Polynomials
- Euclidean Domains
- Principal Ideal Domains
- Unique Factorization Domains
- Gauss's Lemma
- Polynomial Factorization over UFDs
- Eisenstein's Irreducibility Criterion