Definition:Ordered Integral Domain/Definition 2

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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

  • Results about ordered integral domains can be found here.