Definition:Well-Ordered Integral Domain/Definition 2

Definition
Let $\struct {D, +, \times \le}$ be an ordered integral domain whose zero is $0_D$.

$\struct {D, +, \times \le}$ is a well-ordered integral domain every subset $S$ of the set $P$ of (strictly) positive elements of $D$ has a minimal element:
 * $\forall S \subseteq D_{\ge 0_D}: \exists x \in S: \forall a \in S: x \le a$

where $D_{\ge 0_D}$ denotes all the elements $d \in D$ such that $\map P d$.

Also see

 * Equivalence of Definitions of Well-Ordered Integral Domain