Editing Boolean prime ideal theorem (section)

{{short description|Ideals in a Boolean algebra can be extended to prime ideals}}
In [[mathematics]], the '''Boolean prime ideal theorem''' states that [[Ideal (order theory)|ideals]] in a [[Boolean algebra (structure)|Boolean algebra]] can be extended to [[Ideal (order theory)#Prime ideals | prime ideal]]s. A variation of this statement for [[Filter (set theory)|filters on sets]] is known as the [[ultrafilter lemma]]. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, [[Ring (mathematics)|rings]] and prime ideals (of ring theory), or [[distributive lattice]]s and ''maximal'' ideals (of [[order theory]]). This article focuses on prime ideal theorems from order theory.

Although the various prime ideal theorems may appear simple and intuitive, they cannot be deduced in general from the axioms of [[Zermelo–Fraenkel set theory]] without the axiom of choice (abbreviated ZF). Instead, some of the statements turn out to be equivalent to the [[axiom of choice]] (AC), while others—the Boolean prime ideal theorem, for instance—represent a property that is strictly weaker than AC. It is due to this intermediate status between ZF and ZF&nbsp;+&nbsp;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.