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

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.

In a domain, every prime chain below a prime begins at (0)

Statement

Let $R$ be an integral domain and let $p \in Spec (R)$ . Then $(0)$ is a prime ideal of $R$ contained in $p$ . Consequently every strict prime chain below $p$ can be extended downward to a strict chain beginning at $(0)$ .

Facts & Assumptions

Given: An integral domain $R$ and a prime ideal $p \subset R$ .

[L1]

An integral domain is a nonzero commutative ring in which $a b = 0$ implies $a = 0$ or $b = 0$ (Zero divisor, and integral domain: a commutative ring with $1 \neq 0$ and no zero divisors).

[L2]

A prime ideal is a proper ideal $P$ such that $a b \in P$ implies $a \in P$ or $b \in P$ (Prime ideals and maximal ideals in a commutative ring).

Proof

technique · direct

1.1

By [L1], $R$ is nonzero, so $(0) ⊊ R$ . If $a b \in (0)$ , then $a b = 0$ , and [L1] gives $a = 0$ or $b = 0$ . Thus $a \in (0)$ or $b \in (0)$ , so [L2] shows that $(0)$ is prime.

L1L2given

2.1

Every ideal contains $0$ , hence $(0) \subseteq p$ . Therefore any chain of primes below $p$ can be extended by adjoining $(0)$ at the bottom if it is not already present.

step 1.1given

3.1

So every prime chain below $p$ begins at $(0)$ after at most one downward extension.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, §21 (standard reference, not scraped)
Melvin Hochster, Dimension theory and systems of parameters (standard reference, not scraped)