Definition:Ordered Integral Domain/Definition 2

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

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

where $0_R$ is the zero of $R$.

Also see

 * Equivalence of Definitions of Ordered Integral Domain