# Definition:Integral Domain

This page is about Integral Domain in the context of Ring Theory. For other uses, see Domain.

## Definition

### Definition 1

An integral domain $\struct {D, +, \circ}$ is:

a commutative ring which is non-null
with a unity
in which there are no (proper) zero divisors, that is:
$\forall x, y \in D: x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D$

that is, in which all non-zero elements are cancellable.

### Definition 2

An integral domain $\left({D, +, \circ}\right)$ is a commutative ring such that $\left({D^*, \circ}\right)$ is a monoid, all of whose elements are cancellable.

In this context, $D^*$ denotes the ring $D$ without zero: $D \setminus \left\{{0_D}\right\}$.

### Integral Domain Axioms

An integral domain is an algebraic structure $\struct {D, +, \circ}$, on which are defined two binary operations $\circ$ and $+$, which satisfy the following conditions:

 $(\text A 0)$ $:$ Closure under addition $\ds \forall a, b \in D:$ $\ds a + b \in D$ $(\text A 1)$ $:$ Associativity of addition $\ds \forall a, b, c \in D:$ $\ds \paren {a + b} + c = a + \paren {b + c}$ $(\text A 2)$ $:$ Commutativity of addition $\ds \forall a, b \in D:$ $\ds a + b = b + a$ $(\text A 3)$ $:$ Identity element for addition: the zero $\ds \exists 0_D \in D: \forall a \in D:$ $\ds a + 0_D = a = 0_D + a$ $(\text A 4)$ $:$ Inverse elements for addition: negative elements $\ds \forall a \in D: \exists a' \in D:$ $\ds a + a' = 0_D = a' + a$ $(\text M 0)$ $:$ Closure under product $\ds \forall a, b \in D:$ $\ds a \circ b \in D$ $(\text M 1)$ $:$ Associativity of product $\ds \forall a, b, c \in D:$ $\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c}$ $(\text M 2)$ $:$ Commutativity of product $\ds \forall a, b \in D:$ $\ds a \circ b = b \circ a$ $(\text M 3)$ $:$ Identity element for product: the unity $\ds \exists 1_D \in D: \forall a \in D:$ $\ds a \circ 1_D = a = 1_D \circ a$ $(\text D)$ $:$ Product is distributive over addition $\ds \forall a, b, c \in D:$ $\ds a \circ \paren {b + c} = \paren {a \circ b} + \paren {a \circ c}$ $\ds \paren {a + b} \circ c = \paren {a \circ c} + \paren {b \circ c}$ $(\text C)$ $:$ $\struct {D, +, \circ}$ has no (proper) zero divisors $\ds \forall a, b \in D:$ $\ds x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D$

These criteria are called the integral domain axioms.

## Also known as

Some authors refer to this concept as simply a domain.

However, this conflicts with the concept of domain in the context of mappings and relations.

Therefore, it is always best to refer to an integral domain, as to avoid possible confusion.

## Also defined as

Some authors do not require that an integral domain be commutative.

## Also see

• Results about integral domains can be found here.