Editing Ideal (order theory) (section)

==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 '''{{visible anchor|prime ideal}}s'''. Also note that, since we require ideals and filters to be non-empty, every prime ideal is necessarily proper. For lattices, prime ideals can be characterized as follows:

A subset {{mvar|I}} of a lattice <math>(P, \leq)</math> is a prime ideal, if and only if

# {{mvar|I}} is a proper ideal of ''P'', and
# for all elements ''x'' and ''y'' of ''P'', <math>x \wedge y</math> in {{mvar|I}} implies that {{math|''x'' &isin; ''I''}} or {{math|''y'' &isin; ''I''}}.

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

For a [[complete lattice]] the further notion of a '''{{visible anchor|completely prime ideal}}''' is meaningful.
It is defined to be a proper ideal {{mvar|I}} with the additional property that, whenever the meet ([[infimum]]) of some arbitrary set {{math|''A''}} is in {{math|''I''}}, some element of ''A'' is also in {{mvar|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 ZF ([[Zermelo–Fraenkel set theory]] without the [[axiom of choice]]).
This issue is discussed in various [[Boolean prime ideal theorem|prime ideal theorem]]s, which are necessary for many applications that require prime ideals.