# Abstract algebra/Related Articles

From Citizendium, the Citizens' Compendium

*See also changes related to Abstract algebra, or pages that link to Abstract algebra or to this page or whose text contains "Abstract algebra".*

## Contents

## Parent topics

- Algebra [r]: A branch of mathematics concerning the study of structure, relation and quantity.
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## Subtopics

### Disciplines within abstract algebra

- Category theory [r]: Loosely speaking, a class of objects and a collection of morphisms which act upon them; the morphisms can be composed, the composition is associative and there are identity objects and rules of identity.
^{[e]} - Commutative algebra [r]: Branch of mathematics studying commutative rings and related structures.
^{[e]} - Field theory [r]: A subdiscipline of abstract algebra that studies fields, which are mathematical constructs that generalize on the familiar concepts of real number arithmetic.
^{[e]} - Galois theory [r]: Algebra concerned with the relation between solutions of a polynomial equation and the fields containing those solutions.
^{[e]} - Group theory [r]: Branch of mathematics concerned with groups and the description of their properties.
^{[e]} - Linear algebra [r]: Branch of mathematics that deals with the theory of systems of linear equations, matrices, vector spaces, determinants, and linear transformations.
^{[e]} - Ring theory [r]: The mathematical theory of algebraic structures with binary operations of addition and multiplication.
^{[e]} - Universal algebra [r]:
*Add brief definition or description*

### Algebraic structures

- Field [r]: An algebraic structure with operations generalising the familiar concepts of real number arithmetic.
^{[e]} - Group [r]: Set with a binary associative operation such that the operation admits an identity element and each element of the set has an inverse element for the operation.
^{[e]} - Lattice (order) [r]: An ordered set in which any two element set has a supremum and an infimum.
^{[e]} - Module [r]: Mathematical structure of which abelian groups and vector spaces are particular types.
^{[e]} - Monoid [r]: An algebraic structure with an associative binary operation and an identity element.
^{[e]} - Ring [r]: Algebraic structure with two operations, combining an abelian group with a monoid.
^{[e]} - Scheme [r]: Topological space together with commutative rings for all its open sets, which arises from 'glueing together' spectra (spaces of prime ideals) of commutative rings.
^{[e]} - Semigroup [r]: An algebraic structure with an associative binary operation.
^{[e]} - Vector space [r]: A set of vectors that can be added together or scalar multiplied to form new vectors
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- Algebraic geometry [r]: Discipline of mathematics that studies the geometric properties of the objects defined by algebraic equations.
^{[e]} - Algebraic topology [r]:
*Add brief definition or description* - Combinatorics [r]: Branch of mathematics concerning itself, at the elementary level, with counting things.
^{[e]} - Number theory [r]: The study of integers and relations between them.
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