LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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The Noetherian minimal-prime induction split

Statement

Let $R$ be a commutative ring and let $I ⊴ R$ be a proper radical ideal that is not prime. Then there exist elements $x, y \in R ∖ I$ with $x y \in I$ . For any such choice of $x$ and $y$ , every prime ideal minimal over $I$ is minimal over $I + (x)$ or minimal over $I + (y)$ .

Facts & Assumptions

Given: A commutative ring $R$ and a proper radical ideal $I ⊴ R$ that is not prime.

[L1]

A prime ideal is proper and contains one factor whenever it contains a product (Prime ideals and maximal ideals in a commutative ring).

Proof

technique · direct

1.1

Because $I$ is not prime, [L1] gives elements $x, y \in R$ with $x y \in I$ but $x \notin I$ and $y \notin I$ .

L1givenchoose

2.1

Let $p$ be a prime ideal minimal over $I$ . Since $x y \in I \subseteq p$ and $p$ is prime, [L1] gives $x \in p$ or $y \in p$ . If $x \in p$ and $q$ is a prime ideal with $I + (x) \subseteq q \subseteq p$ , then $I \subseteq q \subseteq p$ , so minimality of $p$ over $I$ forces $q = p$ . Thus $p$ is minimal over $I + (x)$ . The same argument with $y$ in place of $x$ shows that if $y \in p$ , then $p$ is minimal over $I + (y)$ .

L1step 1.1given

3.1

Therefore every prime ideal minimal over $I$ appears on one side of the split $I + (x)$ or $I + (y)$ .

step 2.1∎

Depends on

Used by

A radical ideal in a Noetherian ring is a finite intersection of minimal primes Corollary

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Hochster, Introduction to Commutative Algebra, Math 614 notes (2020) (standard reference, not scraped)
The Stacks Project, Section 10.31: Noetherian rings (standard reference, not scraped)