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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

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
Main article: List of group theory topics

Structure

Constructions

Types

Examples

Applications

Ring theory
Main article: Ring theory

General

Structure

Constructions

Types

Examples

Theorems and applications

Field theory
Main article: Field theory (mathematics)

Basic concepts

Types

Applications

Module theory
Main article: Module (mathematics)

General

Structure

Constructions

Types

Concepts and theorems

Representation theory
Main article: List of representation theory topics

Representation theory

Non-associative systems

General

Examples

Generalities

Computer algebra
Main article: computer algebra system

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

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

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