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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Prime quotients of noetherian rings

PDF p. 4: Finite valuative dimension

Statement

If $A$ is a noetherian ring and $p$ is prime, then $A / p$ is a noetherian domain and $\dim (A / p) \leq \dim A$ . Here noetherian means that every ideal is finitely generated.

Facts & Assumptions

Given: A noetherian commutative ring $A$ and a prime ideal $p$ .

Goal: Establish the domain, noetherian and dimension assertions for the quotient.

[F1]

A quotient has Krull dimension at most that of its ring. Dimension decreases under quotients

Proof

1.1givenalgebra

The quotient by a prime ideal is a nonzero domain. If $J$ is an ideal of $A / p$ , its inverse image in $A$ is generated by finitely many elements because $A$ is noetherian; their images generate $J$ . Thus the quotient is noetherian.

2.1F1givenstep 1.1∎

Applying [F1] to $I = p$ gives the dimension inequality.

Depends on

Dimension decreases under quotients

Used by · 1

Finite valuative dimension for noetherian and finite type schemes

Dependency tree · 3 levels

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