CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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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.

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Minimal primes are exactly the primes of height zero

Statement

Let $R$ be a commutative ring and let $p \in Spec (R)$ . Then $p$ is minimal if and only if $ht (p) = 0$ .

Facts & Assumptions

Given: A commutative ring $R$ and a prime ideal $p \subset R$ .

[L1]

The height of $p$ is the supremum of the lengths of strict prime chains ending at $p$ (Height equals local dimension).

Proof

technique · direct

1.1L1given

If $p$ is minimal, there is no strict prime ideal properly contained in $p$ . Therefore every strict chain ending at $p$ has length $0$ , and [L1] gives $ht (p) = 0$ .

1.2L1given

If $ht (p) = 0$ , then [L1] says no strict chain of positive length ends at $p$ . In particular there is no prime ideal properly contained in $p$ , so $p$ is minimal.

2.1step 1.1step 1.2∎

The two implications prove that minimal primes are exactly the primes of height zero.

Depends on

Height equals local dimension

Minimal primes are exactly the primes of height zero

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources