A prime ideal theorem is a theorem stating that every proper ideal is contained in some prime ideal. A prime ideal theorem is typically equivalent to the ultrafilter principle (UF), a weak form of the axiom of choice (AC).
We say ‘a’ prime ideal theorem (PIT) instead of ‘the’ prime ideal theorem, since we have not said what the ideals are in. There are several examples:
The PIT for rings is equivalent to UF.
The PIT for distributive lattices is equivalent to UF.
The PIT for Boolean algebras is equivalent to UF.
The PIT for rigs, which subsumes all of the above, is probably also equivalent to UF; in any case, it follows from AC.
One typically proves a prime ideal theorem with Zorn's Lemma, unless one is specifically trying to use something weaker.
Compare the maximal ideal theorem.