TheoremStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A Noetherian ring has finitely many minimal prime ideals

Statement

Let $R$ be a Noetherian commutative ring. Then $R$ has only finitely many minimal prime ideals.

This theorem inherits only the dependent-choice cost already recorded in the cited Noetherian-induction corollary.

Facts & Assumptions

Given: A Noetherian commutative ring $R$ .

[L1]

The nilradical is $Nil (R) = \sqrt{(0)}$ , and every ideal of the form $\sqrt{I}$ is radical (The nilradical and reduced rings, The radical of an ideal is an ideal).

[L2]

Every radical ideal of a Noetherian ring is a finite intersection of its minimal primes, and hence has only finitely many minimal primes (A radical ideal in a Noetherian ring is a finite intersection of minimal primes).

Proof

technique · direct

1.1

Let $N = Nil (R)$ . By [L1], $N$ is a radical ideal, so [L2] gives only finitely many prime ideals minimal over $N$ .

L1L2

1.2

A prime ideal contains $(0)$ if and only if it contains every nilpotent element, hence if and only if it contains $N$ . Therefore the prime ideals minimal over $(0)$ are exactly the prime ideals minimal over $N$ .

L1givenalgebra

2.1

Combining steps 1.1 and 1.2 shows that $R$ has only finitely many minimal prime ideals.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 The spectrum of a ring (standard reference, not scraped)
A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §13 and §17 (standard reference, not scraped)
The Stacks Project, Section 10.31: Noetherian rings (standard reference, not scraped)