List of abstract algebra topics
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
- Homomorphisms
- *Kernels and cokernels
- *Image and coimage
- *Epimorphisms and monomorphisms
- **Isomorphisms
- ***Isomorphism theorems
- Direct sum
- *Direct limit
- Direct product
- *Inverse limit
- Quotient objects: quotient group, quotient ring, quotient module etc.
- Tensor product
- 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
- *Inverse semigroup
- *Krohn–Rhodes theory
- *Semigroup algebra
- *Transformation semigroup
- Monoid
- *Aperiodic monoid
- *Free monoid
- *Monoid (category theory)
- *Monoid factorisation
- *Syntactic monoid
Group theory
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
- *Trivial group
- *Additive group
- *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
- *Circle group
- *Linear group
- *Orthogonal 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
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
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
Module theory
General
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
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
- *Jordan algebra
- *Alternative algebra
- *Power associativity
- *Flexible algebra
Examples
- Cayley–Dickson construction
- *Octonions
- *Sedenions
- *Trigintaduonions
- 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)