Introduction
Rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. The most familiar example of a ring is the set of all integers .
Contents
Commutative Rings
Ring, Field and Integral Domain
Polynomial Rings
Homomorphisms and Ideals
Principle Ideal Domains
Relations and Maximal Ideals
Factorization
Fields
Fields and Field Extensions
Fields Construction
Finite Fields
Noncommutative Rings and Algebras
Noncommutative Rings
Grading and Filtration
Coalgebra and Hopf Algebra
Modules
Modules
Tensor Products of Modules
Presentation of Finitely Generated Modules