Alphabeta Math
TheoremStatement: Literature-sourcedProof: Literature-sourcedverified 2026-09-24 (gpt-6-sol)
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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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A Noetherian polynomial ring has dimension one larger

Statement

Assume the Axiom of Choice. Let R be a Noetherian commutative ring of finite Krull dimension. Then dim⁡R[x]=dim⁡R+1.

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice) and a Noetherian commutative ring R of finite dimension.

[L1]

Every prime chain in R extends to a prime chain in R[x] that is longer by one step (A prime chain in R extends to a longer chain in R[x]).

[L2]

Under Choice, every prime chain in R[x] has length at most dim⁡R+1 (A prime chain in R[x] has length at most one more than its contraction chain).

Proof

technique · direct
1.1L1given

Fact [L1] gives dim⁡R[x]≥dim⁡R+1.

1.2L2given

Fact [L2] gives dim⁡R[x]≤dim⁡R+1.

2.1step 1.1step 1.2∎

Therefore dim⁡R[x]=dim⁡R+1.

Depends on

Used by

Dependency tree · two levels

14 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