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.
Basic language
Algebraic structures are defined primarily as
sets with operations .
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.
Advanced concepts:
Semigroups and monoids
Group theory
Structure
Constructions
Types
Examples
Applications
Ring theory
General
Structure
Constructions
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
Field theory
Basic concepts
Types
Applications
Module theory
General
Structure
Constructions
Types
Concepts and theorems
Representation theory
Representation theory
Non-associative systems
General
Examples
Generalities
Computer algebra
See also
Areas Basic concepts Algebraic structures Linear and multilinear algebra Algebraic constructions Topic lists Glossaries