List of abstract algebra topics
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in pedagogy.
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
- Finitary operation
Structure preserving maps called homomorphisms are vital in the study of algebraic objects.
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 product
- Quotient objects: quotient group, quotient ring, quotient module etc.
- Tensor product
Advanced concepts:
Semigroups and monoids
- Semigroup
- Subsemigroup
- Free semigroup
- Green's relations
- Inverse semigroup (or inversion semigroup, cf. [1])
- Krohn–Rhodes theory
- Semigroup algebra
- Transformation semigroup
- Monoid
Group theory
Structure
Constructions
Types
- Simple group
- Finite group
- Abelian group
- Cyclic group
- Solvable group
- Nilpotent group
- Divisible group
- Dedekind group, Hamiltonian group
Examples
Applications
Ring theory
General
- Ring (mathematics)
- Commutative algebra, Commutative ring
- Ring theory, Noncommutative ring
- Algebra over a field
- Relatives to rings: Semiring, Nearring, Rig (algebra)
Structure
- Subring, Subalgebra
- Ring ideal
- Jacobson radical
- Socle of a ring
- unit (ring theory), Idempotent, Nilpotent, Zero divisor
- Characteristic (algebra)
- Ring homomorphism, Algebra homomorphism
- Graded algebra
- Morita equivalence
- 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
- 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)
- 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
- Jordan algebra
- Differential algebra
- Banach algebra
Examples
Theorems and applications
- Algebraic geometry
- Hilbert's basis theorem
- Hopkins–Levitzki theorem
- Krull's principal ideal theorem
- Levitzky's theorem
- Galois theory
- Artin-Wedderburn theorem
- Jacobson density theorem
- Wedderburn's little theorem
- Lasker–Noether theorem
Field theory
Basic concepts
- Field (mathematics)
- Subfield (mathematics)
- Field extension
- 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
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:
- Coherent module
- Finitely-generated module
- Finitely-presented module
- Finitely related module
- Algebraically compact module
- Reflexive module
Concepts and theorems
- Composition series
- Structure theorem for finitely generated modules over a principal ideal domain
- Homological 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
Representation theory
Representation theory (& outline)
- Algebra representation
- Group representation
- Lie algebra representation
- Maschke's theorem
- Schur's lemma
- Equivariant map
- Frobenius reciprocity
- 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)
- Nonassociative ring, Non-associative algebra
Examples
Generalities
- Algebraic structure
- Universal algebra
- Variety (universal algebra)
- Congruence relation
- Free object
- Generating set (universal algebra)
- Clone (algebra)
- Kernel of a function
- Universal property
- Filtration (mathematics)
- Category theory
- Monoidal category
- Groupoid
- Group object
- Coalgebra
- Bialgebra
- Hopf algebra
- Magma object
- Torsion (algebra)