# Category:Prime Ideals

This category contains results about Prime Ideals in the context of Order Theory.

Let $I$ be an ideal in an ordered set $S$.

Then $I$ is a **prime ideal** in $S$ if and only if $S \setminus I$ is a filter.

## Pages in category "Prime Ideals"

The following 18 pages are in this category, out of 18 total.

### C

### F

### I

- If Element Does Not Belong to Ideal then There Exists Prime Ideal Including Ideal and Excluding Element
- If Ideal and Filter are Disjoint then There Exists Prime Filter Including Filter and Disjoint from Ideal
- If Ideal and Filter are Disjoint then There Exists Prime Ideal Including Ideal and Disjoint from Filter
- Integral Domain iff Zero Ideal is Prime
- Intersection of Chain of Prime Ideals of Commutative Ring is Prime Ideal