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.
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. ) 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