  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.
