# Category:Definitions/Well-Ordered Integral Domains

This category contains definitions related to Well-Ordered Integral Domains.
Related results can be found in Category:Well-Ordered Integral Domains.

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 $\map P d$.

## Pages in category "Definitions/Well-Ordered Integral Domains"

The following 3 pages are in this category, out of 3 total.