Boolean prime ideal theorem
Basic definition
An ideal in a Boolean algebra A is a subset I of A such that
- Failed to parse (syntax error): {\displaystyle \mbox{for\ all}\ x\in I,\ \mbox{and\ all}\ y\in A\ \mbox{if}\ y\leq x\ \mbox{then}\ y\in I.}
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 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.
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-Frankel 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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