A **ring** is an algebraic structure comprised of a set paired with two operations on the set, which are designated as addition and multiplication . As a group can be conceptualized as an ordered pair of a set and an operation, , a ring can be conceptualized as an ordered triple .

A set with addition and multiplication, , is a field if and only if it satisfies the following properties:

- Commutativity of addition — For all ,
- Associativity of both addition and multiplication — For all , and
- Additive Identity — There exists a "zero" element, , called an additive identity, such that , for all
- Additive Inverses — For each , there exists a , called an additive inverse of , such that
- Distributive property — For all , and
- Closure of addition and multiplication — For all , and

A prototype of these kind of structures in the ring of the integers, .

We will often abbreviate the multiplication of two elements, , by juxtaposition of the elements, . Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, .

We can also denote as the additive inverse of any . Furthermore, we can define an operation, called subtraction by .

## Contents

## Optional Properties

Optionally, a ring may have additional properties:

- We define to be a
**commutative ring**if the multiplication is commutative: for all - We define to be a
**ring with unity**if there exists a multiplicative identity : for all- Furthermore, a commutative ring with unity is a
**field**if every element except 0 has a multiplicative inverse: For each non-zero , there exists a such that

- Furthermore, a commutative ring with unity is a
- We define to be an
**integral domain**if it is commutative, has unity, and the zero product rule holds: implies either or for all .

## Important Results

From the given axioms for a ring , it can be deduced that:

- is an abelian group
- , for all
- , for all
- , for all
- If has unity
- , for all

- Multiplication distributes over subtraction.

## Substructures, morphisms and quotients

- subring
- ideal
- prime ideal
- maximal ideal
- field
- radical
- nilradical
- ring homomorphism
- quotient ring
- fundamental theorems