Algebraic structure

In mathematics an algebraic structure is a set with one, two or more binary operations on it.

The basic algebraic structures with one binary operation are the following:

• Magma (mathematics)

A set with a binary operation.

• Semigroup

A set with an operation which is associative

• Monoid

A semigroup with an identity element

• Group

A monoid where each element has a corresponding inverse element

• Commutative group

A group with a commutative operation

The basic algebraic structures with two binary operations are the following:

• Ring

A set with two operations, often called addition and multiplication. The set with the operation of addition forms a commutative group, and with the operation of multiplication it forms a semigroup (many people define a ring so that the set with multiplication is actually a monoid). Addition and multiplication in a ring satisfy the distributive property

• Commutative ring

A ring whose multiplication is commutative

• Field

A commutative ring where the set with multiplication is a group.

Examples are

• The whole numbers (natural numbers together with zero) with addition (or multiplication) is a monoid, but is not a group

• The integers with addition is a commutative group, but with multiplication is just a monoid

• The integers with addition and multiplication is a commutative ring, but not a field

• The rational numbers, the real numbers and the complex numbers with the ordinary addition and ordinary multiplication are fields.