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Maximal Ideals in Commutative Rings and the Axiom of Choice
- Publication Year :
- 2024
-
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 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.
- Subjects :
- Mathematics - Commutative Algebra
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2404.18351
- Document Type :
- Working Paper