Ideal (order Theory) - Prime Ideals

An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.e. ideals in the inverse order. Such ideals are called prime ideals. Also note that, since we require ideals and filters to be non-empty, every prime filter is necessarily proper. For lattices, prime ideals can be characterized as follows:

A subset I of a lattice (P,≤) is a prime ideal, if and only if

1. I is an ideal of P, and
2. for every elements x and y of P, xy in I implies that x is in I or y is in I.

It is easily checked that this indeed is equivalent to stating that P\I is a filter (which is then also prime, in the dual sense).

For a complete lattice the further notion of a completely prime ideal is meaningful. It is defined to be a proper ideal I with the additional property that, whenever the meet (infimum) of some arbitrary set A is in I, some element of A is also in I. So this is just a specific prime ideal that extends the above conditions to infinite meets.

The existence of prime ideals is in general not obvious, and often a satisfactory amount of prime ideals cannot be derived within Zermelo–Fraenkel set theory. This issue is discussed in various prime ideal theorems, which are necessary for many applications that require prime ideals.