Abstract Algebra

Abstract algebra (a.k.a. modern algebra), is the study of algebraic structures.
== Algebraic structure ==
An algebraic structure consists of a:
# Non-empty set, say X (called the underlying set, carrier set, or domain)
# Collection of operations defined on X
# Finite set of identities, known as axioms, that these operations must satisfy
Common axioms include: commutativity, associativity, and distributivity which are expressed as identities. E.g. an operation \star is commutative if x \star y = y \star x.
== Common algebraic structures ==
=== One set with operations: ===
Simple structures: one set, no binary operation:
* set
Group-like structures: one set, one binary operation:
* Semigroup
* Monoid
* Group
* Abelian group a.k.a commutative group
* Magma
* Quasigroup
* Loop
* Semilattice
Ring-like structures: one set, two binary operations:
* Semi-ring
* Near-ring
* Ring
** Commutative ring
** Division ring
** Field
* Lie Ring
* Jordan Ring
* Boolean Ring
* Lattice
* Boolean Algebra
=== Two sets with operations: ===
* Module
* Vector space
* Algebra over a field
== Branches of abstract algebra ==
=== Group theory ===
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=== Ring theory ===
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=== Field theory ===
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