Formal statement

The Boolean prime ideal theorem is the following statement:

Let B be a Boolean algebra, let I be an (order) ideal and let F be an (order) filter of B, such that I and F are disjoint. Then there is an ideal that contains I, is disjoint with F, and is maximal among the ideals with these properties.

Choosing F = {1}, we obtain a bona-fide maximal ideal containing I. Note that, in a Boolean algebra, the notions maximal ideal and prime ideal coincide.

Consequences and applications

Intuitively, the Boolean prime ideal theorem states that there are "enough" prime ideals in a Boolean algebra in the sense that we can extend every ideal to a maximal one. This is of practical importance for proving Stone's representation theorem for Boolean algebras, known as Stone duality, in which one equips the set of all prime ideals with a certain topology and can indeed regain the original Boolean algebra (up to isomorphism) from this data.

On the other hand, the Boolean prime ideal theorem cannot be proven using only the Zermelo-Fraenkel axioms of set theory. Using the Axiom of Choice in guise of Zorn's lemma is sufficient but not necessary. Indeed, it turns out that the theorem is equivalent to a similar theorem for distributive lattices and that both are strictly weaker than the Axiom of Choice.

Literature