# 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.

## Also presented as

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