Definition:Ordered Integral Domain/Definition 2

Definition

An ordered integral domain is an ordered ring $\struct {D, +, \times, \le}$ which is also an integral domain.

That is, it is an integral domain with an ordering $\le$ compatible with the ring structure of $\struct {D, +, \times}$:

\((\text {OID} 1)\)	$:$		$\le$ is compatible with ring addition:	\(\ds \forall a, b, c \in D:\)		\(\ds a \le b \)	\(\ds \implies \)	\(\ds \paren {a + c} \le \paren {b + c} \)
\((\text {OID} 2)\)	$:$		Strict positivity is closed under ring product:	\(\ds \forall a, b \in D:\)		\(\ds 0_D \le a, 0_D \le b \)	\(\ds \implies \)	\(\ds 0_D \le a \times b \)

where $0_D$ is the zero of $D$.

Also see

• Equivalence of Definitions of Ordered Integral Domain

• Results about ordered integral domains can be found here.