Axiom:Integral Domain Axioms

Definition

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.

These can also be presented as:

$(\text A)$	$:$	$\struct {D, +}$ is an abelian group
$(\text M)$	$:$	$\struct {D, \circ}$ is a commutative monoid
$(\text D)$	$:$	$\circ$ distributes over $+$
$(\text C)$	$:$	$D$ has no (proper) zero divisors

1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 3$. Definition of an Integral Domain

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