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
- I is an ideal of P, and
- 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.
Read more about this topic: Ideal (order Theory)
Famous quotes containing the words prime and/or ideals:
“Vanessa wanted to be a ballerina. Dad had such hopes for her.... Corin was the academically brilliant one, and a fencer of Olympic standard. Everything was expected of them, and they fulfilled all expectations. But I was the one of whom nothing was expected. I remember a game the three of us played. Vanessa was the President of the United States, Corin was the British Prime Minister—and I was the royal dog.”
—Lynn Redgrave (b. 1943)
“Institutional psychiatry is a continuation of the Inquisition. All that has really changed is the vocabulary and the social style. The vocabulary conforms to the intellectual expectations of our age: it is a pseudo-medical jargon that parodies the concepts of science. The social style conforms to the political expectations of our age: it is a pseudo-liberal social movement that parodies the ideals of freedom and rationality.”
—Thomas Szasz (b. 1920)