Boolean prime ideal theorem
From Academic Kids

In mathematics, a number of prime ideal theorems for guaranteeing the existence of certain subsets of an abstract algebra can be stated. Among the most popular statements of this form is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, e.g. rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This article currently focuses on prime ideal theorems from order theory.
Although the various prime ideal theorems may appear simple and intuitive, they can in general not be derived from the axioms of ZermeloFraenkel set theory (ZF). Instead, some of the statements turn out to be equivalent to the axiom of choice (AC), while others, like the Boolean prime ideal theorem, represent a property that is strictly weaker than AC. It is due to this intermediate status between ZF and ZF+AC (ZFC) that the Boolean prime ideal theorem is often taken as an axiom of set theory. The abbreviations BPI or PIT (for Boolean algebras) are sometimes used to refer to this additional axiom.
Contents 
Prime ideal theorems
Before proceeding to actual prime ideal theorems, recall that an order theoretical ideal is just a (nonempty) directed lower set. If the considered poset has binary suprema like the posets within this article, then this is equivalently characterized as a lower set I which is closed for binary suprema (i.e. x, y in I imply x<math>\vee<math>y in I). An ideal I is prime if, whenever an infimum x<math>\wedge<math>y is in I, one also has x in I or y in I. Ideals are proper if they are not equal to the whole poset.
Historically, the first statement relating to later prime ideal theorems was in fact referring to filters  subsets that are ideals with respect to the dual order. The ultrafilter lemma states that every filter on a set is contained within some maximal (proper) filter  an ultrafilter. Recall that filters on sets are just proper filters of the Boolean algebra of its powerset. In this special case, maximal filters (i.e. filters that are not strict subsets of any proper filter) and prime filters (i.e. filters that, with each union of subsets X and Y, also contain X or Y) coincide. Thus the (equivalent) dual of this statement assures that every ideal of a powerset is contained in a prime ideal.
The above statement lead to various generalized prime ideal theorems, each of which exists in a weak and in a strong form. Weak prime ideal theorems state that every nontrivial algebra of a certain class has at least one prime ideal. In contrast, strong prime ideal theorems require that every ideal that is disjoint from a given filter can be extended to a prime ideal which is still disjoint from that filter. In the case of algebras that are not posets, one uses different substructures instead of filters. Many forms of these theorems are actually known to be equivalent, so that the assertion that "PIT" holds is usually taken as the assertion that the corresponding statement for Boolean algebras (BPI) is valid.
This article is mainly dealing with strong prime ideal theorems. It would be helpful to answer the question which weak forms are in fact equivalent to the strong version.
Another variation of similar theorems is obtained by replacing each occurrence of prime ideal by maximal ideal. The corresponding maximal ideal theorems (MIT) are often  though not always  stronger than their PIT equivalents.
Boolean prime ideal theorem
The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras. Thus the formal statement is:
 Let B be a Boolean algebra, let I be an ideal and let F be a filter of B, such that I and F are disjoint. Then I is contained in some prime ideal of B that is disjoint from F.
We refer to this statement as BPI. This situation can be expressed in various different ways. For this purpose, recall the following theorem:
For any ideal I of a Boolean algebra B, the following are equivalent:
 I is a prime ideal.
 I is a maximal proper ideal, i.e. for any proper ideal J, if I is contained in J then I = J.
 For every element a of B, I contains exactly one of {a, ¬a}.
This theorem is a wellknown fact for Boolean algebras. Its dual establishes the equivalence of prime filters and ultrafilters. Note that the last property is in fact selfdual  only the prior assumption that I is an ideal gives the full characterization. It is worth mentioning that all of the implications within this theorem can be proven in classical ZermeloFraenkel set theory.
Thus the following (strong) maximal ideal theorem (MIT) for Boolean algebras is equivalent to BPI:
 Let B be a Boolean algebra, let I be an ideal and let F be a filter of B, such that I and F are disjoint. Then I is contained in some maximal ideal of B that is disjoint from F.
Note that one requires "global" maximality, not just maximality with respect to being disjoint from F. Yet, this variation yields another equivalent characterization of BPI:
 Let B be a Boolean algebra, let I be an ideal and let F be a filter of B, such that I and F are disjoint. Then I is contained in some ideal of B that is maximal among all ideals disjoint from F.
The fact that this statement is equivalent to BPI is easily established by noting the following theorem: For any distributive lattice L, if an ideal I is maximal among all ideals of L that are disjoint to a given filter F, then I is a prime ideal. The proof for this statement (which can again be carried out in ZF set theory) is included in the article on ideals. Since any Boolean algebra is a distributive lattice, this shows the desired implication.
All of the above statements are now easily seen to be equivalent. Going even further, one can exploit the fact the dual orders of Boolean algebras are exactly the Boolean algebras themselves. Hence, when taking the equivalent duals of all former statements, one ends up with a number of theorems that equally apply to Boolean algebras, but where every occurrence of ideal is replaced by filter. It is worth noting that for the special case where the Boolean algebra under consideration is a powerset with the subset ordering, the "maximal filter theorem" is called the ultrafilter lemma. Apparently, the ultrafilter lemma also implies BPI, such that both statements are equivalent  please confirm if this is known to you.
Summing up, for Boolean algebras, the strong MIT, the strong PIT, and the equivalent statements with filters in place of ideals are all equivalent. It is known that all of these statements are consequences of the axiom of choice (the easy proof makes use of Zorn's lemma), but cannot be proven in classical ZermeloFraenkel set theory. Yet, the BPI is strictly weaker than the axiom of choice, though the proof of this statement is rather nontrivial.
Further prime ideal theorems
The prototypical properties that were discussed for Boolean algebras in the above section can easily be modified to include more general lattices, such as distributive lattices or Heyting algebras. However, in these cases maximal ideals are different from prime ideals, and the relation between PITs and MITs is not obvious.
Indeed, it turns out that the MITs for distributive lattices and even for Heyting algebras are equivalent to the axiom of choice. On the other hand, it is known that the strong PIT for distributive lattices is equivalent to BPI (i.e. to the MIT and PIT for Boolean algebras). Hence this statement is strictly weaker than the axiom of choice. Furthermore, observe that Heyting algebras are not self dual, and thus using filters in place of ideals yields different theorems in this setting. Maybe surprisingly, the MIT for the duals of Heyting algebras is not stronger than BPI, which is in sharp contrast to the abovementioned MIT for Heyting algebras.
Finally, prime ideal theorems do also exist for other (not ordertheoretical) abstract algebras. For example, the MIT for rings implies the axiom of choice. This situation requires to replace the ordertheoretic term "filter" by other concepts  for rings a "multiplicatively closed subset" is appropriate.
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, a special case of 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. Furthermore, it turns out that in applications one can freely choose either to work with prime ideals or with prime filters, because every ideal uniquely determines a filter: the set of all Boolean complements of its elements. Both approaches are found in the literature.
Many other theorems of general topology that are often said to rely on the axiom of choice are in fact equivalent to BPI. Feel free to add some examples.
See also
Literature
 B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, 2^{nd} edition, Cambridge University Press, 2002.
 An easy to read introduction, showing the equivalence of PIT for Boolean algebras and distributive lattices.
 P. T. Johnstone, Stone Spaces, Cambridge studies in advanced mathematics 3, Cambridge University Press, 1982.
 The theory in this book often requires choice principles. The notes on various chapters discuss the general relation of the theorems to PIT and MIT for various structures (though mostly lattices) and give pointers to further literature.
 B. Banaschewski, The Power of the Ultrafilter Theorem, Journal of the London Mathematical Society (2) 27, 193202, 1983.
 Discusses the status of the ultrafilter lemma.
 M. Erné, Prime Ideal Theory for General Algebras, Applied Categorical Structures 8, 115144, 2000.
 Gives many equivalent statements for the BPI, including prime ideal theorems for other algebraic structures. PITs are considered as special instances of separation lemmas.