- List of abstract algebra topics
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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 four-group
- 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)
- 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
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)
- 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
- 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
- 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
- Finitely-generated module
- Finitely-presented 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
Non-associative 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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