This is a list of abstract algebra topics.
Contents |
Basic language
- Unary operator
- Binary operation
- Closure (mathematics)
- Arity
- Commutative operation
- Identity element
- Additive inverse, multiplicative inverse, inverse element
- Cancellation property
- Magma object
- Associativity
- Distributivity
Semigroups and monoids
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
- Free group
- Presentation of a group
- Free product
- Word problem for groups
- Burnside's problem
Ring theory
- Ring (mathematics)
- Subring
- Ring homomorphism
- 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
- Hypercomplex number
- Rig (algebra)
- Nearring
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
- Free module
- Finitely-generated module
- Finitely-presented module
- Projective module
- Injective module
- Flat module
- Direct sum
- Simple module
- Indecomposable module
- Pure submodule
- Algebraically compact module
- Bimodule
- Annihilator
- 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 topics
- 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
Generalities
- Algebraic structure
- Universal algebra
- Kernel of a function
- Universal property
- Filtration (mathematics)
- Category theory
- Torsion (algebra)
Computer algebra
- Main article: computer algebra system.
See also
- list of algebraic structures
list of commutative algebra topics list of homological algebra topics - list of linear algebra topics
- glossary of field theory
- glossary of group theory
- glossary of ring theory
- glossary of tensor theory
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