List of abstract algebra topics

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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[edit]

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[edit]

Group theory[edit]

Structure
Constructions
Types
Examples
Applications

Ring theory[edit]

General
Structure
Constructions
Types
Examples
Theorems and applications

Field theory[edit]

Basic concepts
Types
Applications

Module theory[edit]

General
Structure
Constructions
Types
Concepts and theorems

Representation theory[edit]

Representation theory

Non-associative systems[edit]

General
Examples

Generalities[edit]

Computer algebra[edit]

See also[edit]