prime ideal theorem

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. We list some representative examples of prime ideal theorems, all of which are equivalent to (UF) in ZF or even in BZ (bounded Zermelo set theory): The...