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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 (over a fixed ring)
- 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 (or inversion semigroup, cf. [https://www.encyclopediaofmath.org/index.php/Inversion_semi-group])
- 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
- Stable range condition
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
- Wedderburn–Artin 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
- Galois theory
- Galois group
- Inverse Galois problem
- Kummer theory
Module theory
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
Representation theory (& outline)
- 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
- 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)
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