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

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