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.

A prime chain in R extends to a longer chain in R[x]

Statement

Let $R$ be a commutative ring. If $p_{0} ⊊ \dots ⊊ p_{d}$ is a strict prime chain in $R$ , then $p_{0} R [x] ⊊ \dots ⊊ p_{d} R [x] ⊊ p_{d} R [x] + (x)$ is a strict prime chain in $R [x]$ . Consequently $\dim R [x] \geq \dim R + 1$ whenever $\dim R$ is finite.

Facts & Assumptions

Given: A commutative ring $R$ and a strict prime chain $p_{0} ⊊ \dots ⊊ p_{d}$ in $R$ .

[L1]

For a prime ideal $p$ , the quotient $R / p$ is an integral domain ( $R / P$ is an integral domain if and only if $P$ is a prime ideal).

[L2]

Prime ideals of a quotient correspond to prime ideals containing the quotient ideal (Prime ideals of a quotient ring are exactly the prime ideals containing the ideal).

[L3]

Krull dimension is computed by strict prime chains (Krull dimension of a nonzero ring).

Proof

technique · direct

1.1L1L2given

For each $i$ , the quotient $R [x] / p_{i} R [x] ≅ (R / p_{i}) [x]$ is a polynomial ring over the domain $R / p_{i}$ , so [L1] and [L2] show that $p_{i} R [x]$ is prime. Strictness of the original chain makes the extended chain strict.

2.1L1L2step 1.1

The quotient by $p_{d} R [x] + (x)$ is again $R / p_{d}$ , a domain, so [L1] and [L2] show that $p_{d} R [x] + (x)$ is prime and strictly contains $p_{d} R [x]$ .

3.1L3step 1.1step 2.1∎

The displayed chain in $R [x]$ therefore has length $d + 1$ , and [L3] yields $\dim R [x] \geq \dim R + 1$ whenever $\dim R$ is finite.

Depends on

Used by

A Noetherian polynomial ring has dimension one larger Theorem

Dependency tree · two levels

9 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, §§18, 21 (standard reference, not scraped)
Melvin Hochster, Dimension theory and systems of parameters (standard reference, not scraped)