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# Boolean prime ideal theorem

### Basic definition

An ideal in a Boolean algebra A is a subset I of A such that

Superficially this may look different from the concept of an ideal in a ring. However, a Boolean algebra is a ring when multiplication and addition are defined in terms of meet and join in the following way:

It is an exercise to show that this concept of ideal is then no different from the concept of ring ideal known to every student of abstract algebra.

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

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.

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