# Definition:Well-Ordered Integral Domain

## Definition

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

### Definition 1

$\struct {D, +, \times \le}$ is a well-ordered integral domain if and only if the ordering $\le$ is a well-ordering on the set $P$ of (strictly) positive elements of $D$.

### Definition 2

$\struct {D, +, \times \le}$ is a well-ordered integral domain if and only if 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

• Results about well-ordered integral domains can be found here.