 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
Overview
History
Basic language
 Unary operator
 Binary operation
 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
 Coimage
Group theory
Main article: List of group theory topics Group (mathematics)
 Finite group
 Coset
 Lagrange's theorem (group theory)
 Abelian group
 Cyclic group
 Klein fourgroup
 Elementary group theory
 Normal subgroup
 Characteristic subgroup
 Derived group
 Centralizer and normalizer
 Automorphism group
 Sylow theorems
 Group action
 Permutation group
 Symmetric group
 Alternating group
 Point group
 Stabilizer subgroup
 Orbit (group theory)
 Orbitstabilizer 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
Main article: Ring theory Ring (mathematics)
 Subring
 Ring homomorphism
 Ring epimorphism
 Ring monomorphism
 Ring isomorphism
 Ring 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
 Unique factorization domain
 Formal power series
 Characteristic (algebra)
 Boolean algebra (structure), Boolean ring, Boolean homomorphism
 Division ring, division algebra
 Quaternions
 Local ring
 Valuation (mathematics)
 Semilocal ring
 Algebra (ring theory)
 Cohen–Macaulay ring
 Gorenstein ring
 Monoid ring
 Group ring
 Group algebra
 Dual number
 Differential algebra
 Artinian ring
 Noetherian ring
 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 Ralgebras
 Brauer group
 Hypercomplex number
 Rig (algebra)
 Nearring
Field theory
Main article: 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
 Field norm
 Field trace
 Formally real field
 Real closed field
 Tensor product of fields
 Inverse Galois problem
Module theory
Main article: Module (mathematics) Free module
 Finitelygenerated module
 Finitelypresented module
 Projective module
 Swan's theorem
 Quillen–Suslin theorem
 Injective module
 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
Main article: List of representation theory topicsRepresentation 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
Nonassociative systems
 Nonassociative ring
 Associator
 Lie algebra (see also list of Lie group topics and list of representation theory topics)
 Jordan algebra
 Magma (algebra)
 Loop (algebra)
 Quasigroup
 Alternative algebra
 Power associativity
 Cayley–Dickson construction
 Octonions
 Sedenions
Generalities
 Algebraic structure
 Universal algebra
 Kernel of a function
 Universal property
 Filtration (mathematics)
 Category theory
 Monoidal category
 Groupoid
 Group object
 Coalgebra
 Bialgebra
 Hopf algebra
 Magma object
 Torsion (algebra)
Computer algebra
Main article: computer algebra system Symbolic mathematics
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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 Abstract algebra
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