A maximal ideal theorem is a theorem stating that every proper ideal is contained in some maximal ideal. A maximal ideal theorem is typically equivalent to the axiom of choice (AC), or a slightly weaker version such as the ultrafilter principle (UF).
We say ‘a’ maximal ideal theorem (MIT) instead of ‘the’ maximal ideal theorem, since we have not said what the ideals are in. There are several examples: * The MIT for rings is equivalent to AC. * The MIT for Heyting algebras is equivalent to AC. * The MIT for Boolean algebras is equivalent to UF. * The MIT for rigs, which subsumes all of the above, is equivalent to AC.
One typically proves a maximal ideal theorem with Zorn's Lemma, unless one is specifically trying to use something weaker.
Compare the prime ideal theorem.