2024/04/29 by Alexei Entin, Entin, Alexei
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2404.18351
openalex publication_date 2024/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ⇒ AC was first proved by Hodges, with subsequent proofs given by Banaschewski and Erné. Here we give another derivation of MIT ⇒ 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.