Definition:Integral Domain

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

Definition

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

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.

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: $D \setminus \left\{{0_D}\right\}$.

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.

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.

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

\((\text A 0)\)	$:$	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	\(\ds \forall a, b \in D:\)	\(\ds x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D \)