List of abstract algebra topics

This is a list of abstract algebra topics, by Wikipedia page. See also:

• list of commutative algebra topics
• list of group theory topics
• list of homological algebra topics
• list of linear algebra topics.
• glossary of field theory
• glossary of group theory
• glossary of ring theory
• glossary of tensor theory.

Basic language

• Unary operator
• Binary operation
  • Multiplication table
• Closure (mathematics)
• Arity
• Commutative operation
• Identity element
• Additive inverse, multiplicative inverse, inverse element
• Cancellation property
• Magma object
• Associativity
• Distributivity

Semigroups and monoids

• Semigroup
  • Free semigroup
  • Green's relations
  • Krohn-Rhodes theory
  • Transformation semigroup
  • Semigroupoid
  • Subsemigroup
• monoid
  • aperiodic monoid
• Coimage

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
  • Rank of an abelian group
  • Divisible group
  • Locally cyclic 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 (group theory)
  • 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 epimorphism
  • Ring monomorphism
  • Ring isomorphism
• Ring ideal
  • Principal ideal
  • Ideal quotient
  • Maximal ideal, prime ideal
  • Jacobson radical
  • Radical of an ideal
• Simple ring
• Product of rings
• Unit (ring theory)
• Order (ring theory)
• Idempotent
• Nilpotent, reduced (ring theory)
• Zero divisor
• Integral domain
• Euclidean domain
• Principal ideal domain
• Domain (ring theory)
• Prime ring
• Primitive ring
• Primitive ideal
• Semiprimitive ring
• Matrix ring
• Jacobson density theorem
• Field (mathematics)
• Field of quotients
• Commutative algebra
• Polynomial, monomial
• Polynomial ring, symmetric algebra
• Center (algebra)
• Integral closure
• Dedekind domain
  • Ideal class group
• Unique factorization domain
• Formal power series
• Characteristic
• Boolean algebra, Boolean ring, Boolean homomorphism
• Division ring, division algebra
• Quaternions
  • Hurwitz quaternion
• Local ring
  • Discrete valuation ring
  • Regular local ring
• Valuation (mathematics)
• Semi-local ring
• Algebra (ring theory)
• Cohen-Macaulay ring
• Gorenstein ring
• Monoid ring
• Group ring
• Group algebra
• Dual number
• Derivation (abstract algebra)
• Artinian ring
• Noetherian ring
  • Hilbert's basis theorem
• Von Neumann regular ring
• Banach algebra
• Associative algebra
  • Subalgebra
  • Algebra over a field
  • Simple algebra
  • Central simple algebra
  • Artin-Wedderburn theorem
  • Skolem-Noether theorem
  • Tensor algebra, free algebra
  • Tensor product of R-algebras
  • Brauer group
• Hypercomplex number
  • Graded algebra
• Rig (algebra)

Field theory

• Subfield (mathematics)
• Field extension
• Multiplicative group
• Algebraic number field
• Global field
• Local field
• Finite field
• Algebraic element
• Algebraic extension
• Symmetric function
• Splitting field
• Separable polynomial
• Separable extension
• Primitive element (field theory)
• Algebraic closure
• Algebraically closed field
• Transcendence degree
• Normal extension
• Galois extension
• Galois theory
  • Galois group
  • Conjugate element (field theory)
  • Abelian extension
  • Kummer theory
• Field norm
• Field trace
• Formally real field
• Real closed field
• Tensor product of fields
• Inverse Galois problem

Module theory

• Free module
• Finitely-generated module
• Finitely-presented module
• Projective module
  • Swan's theorem
  • Quillen-Suslin theorem
• Injective module
  • Injective hull
• Flat module
• Direct sum
• Simple module
• Indecomposable module
• Pure submodule
• Algebraically compact module
• Bimodule
• Annihilator
• Regular sequence (algebra), depth (algebra)
• Localization of a module
• Fractional ideal
• Length of a module
• Fitting lemma
• Artinian module

Representation theory

• Representation of an algebra
• Group representation
  • Regular representation
  • Restricted representation
  • Induced representation
• Maschke's theorem
• Schur's lemma
• Intertwining map
• Affine representation
• Projective representation
• Modular representation theory
• Quiver (mathematics)
• Representation of a Hopf algebra
• Algebra representation of a Hopf algebra
• Algebra representation of a Lie algebra
• Algebra representation of a Lie superalgebra

Non-associative systems

• Nonassociative ring
• Associator
• Lie algebra (see also list of Lie group topics)
  • Jacobi identity
  • Superalgebra
  • Representation of Lie algebras
  • Universal enveloping algebra
  • Virasoro algebra
• Jordan 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 of a function
  • Kernel (algebra)
  • Isomorphism class
  • Isomorphism theorem
  • Fundamental theorem on homomorphisms
• Universal property
• Filtration (abstract algebra)
• Category theory
  • Algebra/set analogy
  • Groupoid
  • Group object
  • Coalgebra
  • Bialgebra
  • Hopf algebra

Computer algebra

See Computer_algebra_system#List_of_Computer_algebra_systems for a list.

• Symbolic mathematics
  • Finite field arithmetic
  • Gröbner basis
  • Buchberger's algorithm
• Axiom computer algebra system, Axiom (algebra software)
• GAP computer algebra system
• Ginac computer algebra system
• Macaulay computer algebra system
• Magma computer algebra system
• Maple computer algebra system
• Mathematica
• Maxima computer algebra system
• PARI-GP computer algebra system
• Reduce computer algebra system
• SINGULAR computer algebra system
• Yacas