Maximal Ideals in Commutative Rings and the Axiom of Choice

TL;DR

This paper explores the logical relationship between the Axiom of Choice and the Maximal Ideal Theorem in set theory and commutative algebra, providing a clear derivation of how MIT implies AC.

Contribution

It offers a new, self-contained proof that the Maximal Ideal Theorem implies the Axiom of Choice, accessible to non-experts in set and ring theory.

Findings

MIT implies AC in ZF set theory.

Provides an accessible derivation of the implication.

Clarifies the logical dependencies between algebraic and set-theoretic principles.

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.