Maximal Ideals in Commutative Rings and the Axiom of Choice

Abstract

It is well-known that within Zermelo-Fraenkel set theory (ZF), the Axiom of Choice (AC) implies the Maximal Ideal Theorem (MIT), namely that every nontrivial commutative ring has a maximal ideal. The converse implication MIT $\Rightarrow$ AC was first proved by Hodges, with subsequent proofs given by Banaschewski and Ern\'e. Here we give another derivation of MIT $\Rightarrow$ AC, aiming to make the exposition self-contained and accessible to non-experts with only introductory familiarity with commutative ring theory and naive set theory.