An ideal in a Boolean algebra A is a subset I of A such that
In Boolean algebras, unlike rings in general, there is no difference between a prime ideal and a maximal ideal.
Both principal and non-principal ideals exist in Boolean algebras.
The Boolean prime ideal theorem states that in a Boolean algebra, every ideal can be extended to a maximal ideal, i.e., to a prime ideal. It is just a special case of a theorem applying more generally to rings, proved by the same sort of application of Zorn's lemma. Why then, does it warrant an article all to itself? Because within the Zermelo-Fraenkel axioms of set theory, it is strictly weaker than the well-known theorem of algebra of which it is but a special case, and mathematical logicians have taken an interest in showing that it is formally equivalent to various other propositions in mathematics.