Ring Theory

This book is meant to be used as a first course in ring theory. No pre-requisite, other then an elementary knowledge of mathematics, is necessary. Rigorous proofs are provided for nearly every result.

Table of contents

Introduction

1. • Rings
   • Properties of rings
   • Integral domains and Fields
   • Subrings
   • Idempotent and Nilpotent elements
   • Characteristic of a ring
   • Ideals in a ring
   • Simple ring
2. Homomorphisms
   • Homomorphisms of rings
   • Theorems on homomorphisms
   • Quotient rings
   • Imbedding of rings
   • Maximal and Prime Ideals
   • Divisibility in rings
3. Some special domains
   • Principal Ideal Domains
   • Euclidean domains
   • Polynomial rings
   • Unique Factorization domain
   • Primitive and irreducible polynomials
   • R(x) as a UFD
4. Extension fields
   • Extension fields
   • Field adjunctions
   • Algebraic elements and algebraic extensions
   • Roots of polynomials
   • Constructions by ruler and compass

Resources

• http://www.polya.uklinux.net/rings.pdf