# Definition: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:

 $(A0)$ $:$ Closure under addition $\displaystyle \forall a, b \in D:$ $\displaystyle a * b \in D$ $(A1)$ $:$ Associativity of addition $\displaystyle \forall a, b, c \in D:$ $\displaystyle \paren {a * b} * c = a * \paren {b * c}$ $(A2)$ $:$ Commutativity of addition $\displaystyle \forall a, b \in D:$ $\displaystyle a * b = b * a$ $(A3)$ $:$ Identity element for addition: the zero $\displaystyle \exists 0_D \in D: \forall a \in D:$ $\displaystyle a * 0_D = a = 0_D * a$ $(A4)$ $:$ Inverse elements for addition: negative elements $\displaystyle \forall a \in D: \exists a' \in D:$ $\displaystyle a * a' = 0_D = a' * a$ $(M0)$ $:$ Closure under product $\displaystyle \forall a, b \in D:$ $\displaystyle a \circ b \in D$ $(M1)$ $:$ Associativity of product $\displaystyle \forall a, b, c \in D:$ $\displaystyle \paren {a \circ b} \circ c = a \circ \paren {b \circ c}$ $(D)$ $:$ Product is distributive over addition $\displaystyle \forall a, b, c \in D:$ $\displaystyle a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}$ $\displaystyle \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}$ $(C)$ $:$ Product is commutative $\displaystyle \forall a, b \in D:$ $\displaystyle a \circ b = b \circ a$ $(U)$ $:$ Identity element for product: the unity $\displaystyle \exists 1_D \in D: \forall a \in D:$ $\displaystyle a \circ 1_D = a = 1_D \circ a$ $(ZD)$ $:$ No proper zero divisors $\displaystyle \forall a, b \in D:$ $\displaystyle a \circ b = 0_D \iff a = 0 \lor b = 0$

These criteria are called the integral domain axioms.

These can be otherwise presented as:

 $(A)$ $:$ $\struct {D, *}$ is an abelian group $(M)$ $:$ $\struct {D \setminus \set 0, \circ}$ is a monoid $(D)$ $:$ $\circ$ distributes over $*$ $(C)$ $:$ $\circ$ is commutative