LemmaStatement: AI-generatedProof: AI-generated

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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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Dimension decreases under quotients

PDF p. 4: Finite valuative dimension

Statement

For a commutative ring $A$ and ideal $I$ , $\dim (A / I) \leq \dim A$ , using $- 1$ for the zero ring.

Facts & Assumptions

Given: A commutative ring $A$ and an ideal $I$ .

Goal: Compare the Krull dimensions of $A / I$ and $A$ .

[F1]

Krull dimension is the supremum of the lengths of strict chains of prime ideals. Valuative dimension and degreewise bounded towers

Proof

1.1givenalgebra

The inverse image of a prime ideal under the quotient map $A \to A / I$ is prime. Distinct ideals of $A / I$ have distinct inverse images, since the map is surjective. Thus every strict prime chain in $A / I$ pulls back to a strict prime chain of the same length in $A$ .

2.1F1step 1.1∎

Taking suprema gives the inequality. If $A / I$ is zero, its dimension is $- 1$ , and the same inequality holds by the convention in [F1].

Depends on

Valuative dimension and degreewise bounded towers

Used by · 2

Dependency tree · 3 levels

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