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.

Basic language

Algebraic structures are defined primarily as sets with operations.

Algebraic structure
- Subobjects: subgroup, subring, subalgebra, submodule etc.
Binary operation
- Closure of an operation
- Associative property
- Distributive property
- Commutative property
Unary operator
- Additive inverse, multiplicative inverse, inverse element
  - Identity element
  - Cancellation property
Finitary operation
- Arity

Structure preserving maps called homomorphisms are vital in the study of algebraic objects.

Homomorphisms
- Kernels and cokernels
- Image and coimage
- Epimorphisms and monomorphisms
  - Isomorphisms
    - Isomorphism theorems

There are several basic ways to combine algebraic objects of the same type to produce a third object of the same type. These constructions are used throughout algebra.

Direct sum
- Direct limit
Direct product
- Inverse limit
Quotient objects: quotient group, quotient ring, quotient module etc.
Tensor product

Advanced concepts:

Category theory
- Category of groups
  - Category of abelian groups
- Category of rings
- Category of modules
  - Morita equivalence, Morita duality
- Category of vector spaces
Homological algebra
- Filtration (algebra)
- Exact sequence
- Functor
Zorn's lemma

Semigroups and monoids

Semigroup
- Subsemigroup
- Free semigroup
- Green's relations
- Inversion semigroup (cf. [1])
- Krohn–Rhodes theory
- Semigroup algebra
- Transformation semigroup
Monoid
- Aperiodic monoid
- Free monoid
- Monoid (category theory)
- Monoid factorisation
- Syntactic monoid

Group theory
Main article: List of group theory topics

Structure

Group (mathematics)
- Lagrange's theorem (group theory)
- Subgroup
  - Coset
  - Normal subgroup
  - Characteristic subgroup
  - Centralizer and normalizer subgroups
  - Derived group
  - Frattini subgroup
  - Fitting subgroup
- Classification of finite simple groups
- Sylow theorems
- Local analysis

Constructions

Free group
- Presentation of a group
- Word problem for groups
Quotient group
- Extension problem
Direct sum, direct product
Semidirect product
- Wreath product

Types

Simple group
Finite group
Abelian group
- Torsion subgroup
- Free abelian group
- Finitely generated abelian group
- Rank of an abelian group
Cyclic group
- Locally cyclic group
Solvable group
- Composition series
Nilpotent group
Divisible group
Dedekind group, Hamiltonian group

Examples

Examples of groups
- Permutation group
- Symmetric group
- Alternating group
- p-group
- List of small groups
- Klein four-group
- Quaternion group
- Dihedral group
- Dicyclic group
- Automorphism group
- Point group

Applications

Group action
- Conjugacy class
- Inner automorphism
- Conjugate closure
- Stabilizer subgroup
- Orbit (group theory)
- Orbit-stabilizer theorem
- Cayley's theorem
- Burnside's lemma
Burnside's problem
Loop group
Fundamental group

Ring theory
Main article: Ring theory

General

Ring (mathematics)
Commutative algebra, Commutative ring
Ring theory, Noncommutative ring
Algebra over a field
- Non-associative algebra
Relatives to rings: Semiring, Nearring, Rig (algebra)

Structure

Subring, Subalgebra
- Center (algebra)
Ring ideal
- Principal ideal
- Ideal quotient
- Maximal ideal, minimal ideal
- Primitive ideal, prime ideal, semiprime ideal
- Radical of an ideal
Jacobson radical
Socle of a ring
unit (ring theory), Idempotent, Nilpotent, Zero divisor
Characteristic (algebra)
Ring homomorphism, Algebra homomorphism
- Ring epimorphism
- Ring monomorphism
- Ring isomorphism
  - Skolem–Noether theorem
Graded algebra
Morita equivalence
- Brauer group

Constructions

Direct sum of rings, Product of rings
Quotient ring
Matrix ring
Endomorphism ring
Polynomial ring
Formal power series
Monoid ring, Group ring
Localization of a ring
Tensor algebra
- Symmetric algebra, Exterior algebra, Clifford algebra
Free algebra
Completion (ring theory)

Types

Field (mathematics), Division ring, division algebra
Simple ring, Central simple algebra, Semisimple ring, Semisimple algebra
Primitive ring, Semiprimitive ring
Prime ring, Semiprime ring, Reduced ring
Integral domain, Domain (ring theory)
- Field of fractions, Integral closure
- Euclidean domain, Principal ideal domain, Unique factorization domain, Dedekind domain, Prüfer domain
Von Neumann regular ring
Quasi-Frobenius ring
Hereditary ring, Semihereditary ring
Local ring, Semi-local ring
Discrete valuation ring
Regular local ring
Cohen–Macaulay ring
Gorenstein ring
Artinian ring, Noetherian ring
Perfect ring, semiperfect ring
Baer ring, Rickart ring
Lie ring, Lie algebra
- Ideal (Lie algebra)
Jordan algebra
Differential algebra
Banach algebra

Examples

Rational number, Real number, Complex number, Quaternions, Octonions
- Hurwitz quaternion
Gaussian integer

Theorems and applications

Algebraic geometry
- Hilbert's Nullstellensatz
Hilbert's basis theorem
Hopkins–Levitzki theorem
Krull's principal ideal theorem
Levitzky's theorem
Galois theory
- Abel–Ruffini theorem
Artin-Wedderburn theorem
Jacobson density theorem
Wedderburn's little theorem
Lasker–Noether theorem

Field theory
Main article: Field theory (mathematics)

Basic concepts

Field (mathematics)
Subfield (mathematics)
- Multiplicative group
  - Primitive element (field theory)
Field extension
- Algebraic extension
  - Splitting field
  - Algebraically closed field
    - Algebraic element
    - Algebraic closure
- Separable extension
  - Separable polynomial
- Normal extension
- Galois extension
- Abelian extension
- Transcendence degree
Field norm
Field trace
Conjugate element (field theory)
Tensor product of fields

Types

Algebraic number field
Global field
Local field
Finite field
Symmetric function
Formally real field
Real closed field

Applications

Galois theory
- Galois group
- Inverse Galois problem
Kummer theory

Module theory
Main article: Module (mathematics)

General

Module (mathematics)
Bimodule
Annihilator (ring theory)

Structure

Submodule
- Pure submodule
Module homomorphism
- Essential submodule
- Superfluous submodule
- Singular submodule
Socle of a module
Radical of a module

Constructions

Free module
Quotient module
Direct sum, Direct product of modules
Direct limit, Inverse limit
Localization of a module
Completion (ring theory)

Types

Simple module, Semisimple module
Indecomposable module
Artinian module, Noetherian module
Homological types:
- Projective module
  - Projective cover
  - Swan's theorem
  - Quillen–Suslin theorem
- Injective module
  - Injective hull
- Flat module
  - Flat cover
Coherent module
Finitely-generated module
Finitely-presented module
Finitely related module
Algebraically compact module
Reflexive module

Concepts and theorems

Composition series
- Length of a module
Structure theorem for finitely generated modules over a principal ideal domain
Homological dimension
- Projective dimension
- Injective dimension
- Flat dimension
- Global dimension
- Weak global dimension
- Cohomological dimension
Krull dimension
Regular sequence (algebra), depth (algebra)
Fitting lemma
Schur's lemma
Nakayama's lemma
Krull–Schmidt theorem
Steinitz exchange lemma
Jordan–Hölder theorem
Artin–Rees lemma
Schanuel's lemma
Morita equivalence
- Progenerator

Representation theory
Main article: List of representation theory topics

Representation theory

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

Non-associative systems

General

Associative property, Associator
Heap (mathematics)
Magma (algebra)
- Loop (algebra), Quasigroup
Nonassociative ring, Non-associative algebra
- Universal enveloping algebra
- Lie algebra (see also list of Lie group topics and list of representation theory topics)
- Jordan algebra
- Alternative algebra
- Power associativity
- Flexible algebra

Examples

Cayley–Dickson construction
- Octonions
- Sedenions
Hyperbolic quaternions
Virasoro algebra

Generalities

Algebraic structure
Universal algebra
- Variety (universal algebra)
- Congruence relation
- Free object
- Generating set (universal algebra)
- 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
Main article: computer algebra system

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