List of abstract algebra topics

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra. The distinction is rarely made in more recent writings.

Contents 1 Overview 2 History 3 Basic language 4 Semigroups and monoids 5 Group theory 6 Ring theory 7 Field theory 8 Module theory 9 Representation theory 10 Non-associative systems 11 Generalities 12 Computer algebra 13 See also

Overview

History

Basic language

• Unary operator
• Binary operation
  • Multiplication table
• Closure (mathematics)
• Arity
• Commutative operation
• Identity element
• Additive inverse, multiplicative inverse, inverse element
• Cancellation property
• 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 (group theory)
• 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 theorems
  • Local analysis
• Group action
  • Permutation group
  • Symmetric group
  • Alternating group
  • Point 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
• Quotient ring
• Simple ring
• Product of rings
• Unit (ring theory)
• Order (ring theory)
• Idempotent
• Nilpotent, reduced ring
• 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 fractions
• Commutative algebra
• Polynomial, monomial
• Polynomial ring, symmetric algebra
• Center (algebra)
• Integral closure
• Dedekind domain
  • Ideal class group
• Unique factorization domain
• Formal power series
• Characteristic (algebra)
• Boolean algebra (structure), 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
• Differential 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)
• Nearring

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 (ring theory)
• Regular sequence (algebra), depth (algebra)
• Localization of a module
• Fractional ideal
• Length of a module
• Fitting lemma
• Artinian module

Representation theory

Representation theory

• Algebra representation
• Group representation
• Lie algebra representation
• Maschke's theorem
• Schur's lemma
• Equivariant map
• Affine representation
• Projective representation
• Modular representation theory
• Quiver (mathematics)
• Representation theory of Hopf algebras

Non-associative systems

• Nonassociative ring
• Associator
• Lie algebra (see also list of Lie group topics and list of representation theory topics)
  • Jacobi identity
  • Superalgebra
  • 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
  • Clone (algebra)
• Kernel of a function
  • Kernel (algebra)
  • Isomorphism class
  • Isomorphism theorem
  • Fundamental theorem on homomorphisms
• Universal property
• Filtration (mathematics)
• Category theory
  • Monoidal category
  • Groupoid
  • Group object
  • Coalgebra
  • Bialgebra
  • Hopf algebra
  • Magma object
• Torsion (algebra)

Computer algebra

• Symbolic mathematics
  • Finite field arithmetic
  • Gröbner basis
  • Buchberger's algorithm

See also

• List of commutative algebra topics
• List of homological algebra topics
• List of linear algebra topics
• List of algebraic structures
• Glossary of field theory
• Glossary of group theory
• Glossary of ring theory
• Glossary of tensor theory

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