Algebraic structure

In mathematics an algebraic structure is a set with one, two, or more binary operations on it. The binary operation takes two elements of the set as inputs, and gives one element of the set as an output.

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.