Definition:Integral Domain Axioms

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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             


Also see


Sources