List of abstract algebra topics
From Wikipedia, the free encyclopedia
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.
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[edit] Basic language
- Unary operator
- Binary operation
- Closure (mathematics)
- Arity
- Commutative operation
- Identity element
- Additive inverse, multiplicative inverse, inverse element
- Cancellation property
- Associativity
- Distributivity
[edit] Semigroups and monoids
[edit] 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
[edit] Ring theory
Main article: 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
[edit] Field theory
Main article: 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
[edit] Module theory
Main article: Module (mathematics)
- 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 (ring theory)
- Regular sequence (algebra), depth (algebra)
- Localization of a module
- Fractional ideal
- Length of a module
- Fitting lemma
- Artinian module
[edit] 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
- Affine representation
- Projective representation
- Modular representation theory
- Quiver (mathematics)
- Representation theory of Hopf algebras
[edit] 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
[edit] Generalities
- Algebraic structure
- Universal algebra
- Kernel of a function
- Universal property
- Filtration (mathematics)
- Category theory
- Torsion (algebra)
[edit] Computer algebra
Main article: computer algebra system
[edit] 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
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