List of abstract algebra topics

This is a list of abstract algebra topics, by Wikipedia page. See also the list of group theory topics, glossary of group theory, glossary of ring theory, glossary of field theory, glossary of tensor theory; and the list of linear algebra topics.

Table of contents

1 Basic language 2 Group theory 3 Ring theory 4 Field theory 5 Module theory 6 Non-associative systems 7 Generalities 8 Computer algebra

Basic language

• Binary operation
  • Multiplication table
• Commutative operation
• Identity element
• Additive inverse, multiplicative inverse
• Associativity
• Distributivity
  • Distributive property
• Semigroup, monoid

Group theory

• Group (mathematics)
• Finite group
• Coset
• Lagrange's theorem
• Abelian group
  • Torsion subgroup
  • Direct sum of groups
  • Free abelian group
  • Finitely generated abelian group
• Cyclic group
• Klein four-group
• Elementary group theory
  • Quaternion group
  • List of small groups
  • Examples of groups
• Normal subgroup
  • Conjugacy class
    • Inner automorphism
    • Conjugate closure
  • Semidirect product
  • Dihedral group
  • Dicyclic group
  • Extension problem
  • Hamiltonian group
  • Simple group
    • Classification of finite simple groups
  • Composition series
  • Solvable group
  • Nilpotent group
• Characteristic subgroup
• Derived group
• Centralizer and normalizer
• Automorphism group
• Sylow theorem
  • Local analysis
• Group action
  • Permutation group
  • Symmetric group
  • Alternating group
  • Stabilizer subgroup
  • Orbit (mathematics)
  • Orbit-stabilizer theorem
  • Cayley's theorem
  • Burnside's lemma
  • Wreath product
• Free group
• Presentation of a group
• Free product
• Word problem for groups
• Burnside's problem

Ring theory

• Ring (mathematics)
• Subring
• Ring homomorphism
• Ring ideal
  • Principal ideal
  • Maximal ideal, prime ideal
  • Jacobson radical
  • Radical of an ideal
• Product of rings
• Idempotent
• Nilpotent
• Zero divisor
• Integral domain
• Euclidean domain
• Principal ideal domain
• Field (mathematics)
• Field of quotients
• Commutative algebra
• Polynomial, monomial
• Center (algebra)
• Integral closure
• Dedekind domain
  • Ideal class group
• Unique factorization domain
• Formal power series
• Characteristic
• Boolean algebra, Boolean ring
• Division ring, division algebra
• Local ring
• Valuation (mathematics)
• Monoid ring
• Group ring
• Dual number
• Derivation
• Artinian
• Noetherian
  • Hilbert's basis theorem
• Banach algebra
• Associative algebra
  • Algebra over a field
  • Tensor algebra
• Hypercomplex number
  • Graded algebra
• Rig (algebra)

Field theory

• Subfield (mathematics)
• Field extension
• Algebraic number field
• Global field
• Local field
• Finite field
• Algebraic extension
• Symmetric function
• Splitting field
• Separable polynomial
• Separable extension
• Primitive element (field theory)
• Algebraic closure
• Algebraically closed field
• Transcendence degree
• Galois theory
  • Galois group
  • Abelian extension
• Tensor product of fields

Module theory

• Free module
• Projective module
  • Swan's theorem
• Finitely-generated module
• Direct sum
• Simple module
• Annihilator
• Representation theory
  • Group representation
    • Regular representation
    • Induced representation
  • Intertwining map
  • Affine representation
  • Projective representation
• Quiver (mathematics)
• Homological algebra
  • Cokernel
  • Exact sequence
  • Chain complex
  • Five lemma
  • Short five lemma
  • Snake lemma
  • Extension (algebra)
  • Central extension
  • Splitting lemma
  • Whitehead problem

Non-associative systems

• Lie algebra
  • Jacobi identity
  • Superalgebra
  • Representation of Lie algebras
  • Universal enveloping algebra
  • Virasoro algebra
• Magma (algebra)
• Loop (algebra)
• Quasigroup
• Alternative algebra
• Power associativity
• Cayley-Dickson construction
  • Octonions
  • Sedenions

Generalities

• Algebraic structure
• Universal algebra
  • Mathematical variety
  • Congruence relation
  • Free object
  • Generating set
• Kernel (function)
  • Kernel (algebra)
  • Isomorphism theorem
  • Fundamental theorem on homomorphisms
• Universal property
• Category theory
  • Algebra/set analogy
  • Groupoid
  • Group object
  • Coalgebra
  • Bialgebra
  • Hopf algebra

Computer algebra

• Grobner basis
• Macaulay computer algebra system
• Magma computer algebra system
• PARI-GP computer algebra system
• SINGULAR computer algebra system