Maximal Ideals in Commutative Rings and the Axiom of Choice
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913915189854208 |
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| author | Entin, Alexei |
| author_facet | Entin, Alexei |
| contents | 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 Erné. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal Ideals in Commutative Rings and the Axiom of Choice Entin, Alexei Commutative Algebra 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 Erné. 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. |
| title | Maximal Ideals in Commutative Rings and the Axiom of Choice |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2404.18351 |