by Y Fan (Wuhan University, P R China) , Q Y Xiong (Wuhan University, P R China) , & Y L Zheng (Wuhan University, P R China)

This volume is based on the lectures given by the authors at Wuhan University and Hubei University in courses on abstract algebra. It presents the fundamental concepts and basic properties of groups, rings, modules and fields, including the interplay between them and other mathematical branches and applied aspects.

• Preliminaries:
  • Sets
  • Logic
  • Relations
  • Maps
  • Zorn's Lemma
• Groups:
  • Transformations and Permutations
  • Groups
  • Subgroups
  • Homomorphisms, Isomorphisms
  • Cosets
  • Normal Subgroups, Quotient Groups
  • Homomorphism Theorems
  • Cyclic Groups, Orders of Elements
  • Direct Products
• Rings:
  • Fundamentals
  • Zero Divisors, Invertible Elements
  • Ideals, Residue Rings
  • Homomorphism Theorems
  • Prime Ideals, Maximal Ideals
  • Direct Sums
  • Fraction Fields of Integral Domains
  • Polynomial Rings
  • Factorial Rings
  • Polynomial Rings over Factorial Rings
• Modules:
  • Modules
  • Homomorphisms
  • Direct Products, Direct Sums
  • Exact Sequences of Homomorphisms
  • Free Modules, Matrices over Rings
  • Matrices over Division Rings
  • Matrices over Commutative Rings
  • Algebras over Commutative Rings
  • Tensor Products
  • Projective Modules, Injective Modules
• Fields:
  • Subfields and Extensions
  • Single Extensions
  • Algebraic Extensions
  • Splitting Fields, Normal Extensions
  • Two Applications
  • Separability, Multiple Roots
  • Finite Fields
  • Coding
  • p-adic Numbers
  • Quaternions