Alphabeta Math
TheoremStatement: Literature-sourcedProof: Literature-sourced
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.

Noetherian domains have equal Krull and valuative dimensions

PDF p. 4: Finite valuative dimension

Statement

For a noetherian integral domain D, dim⁡vD=dim⁡D, allowing infinity. Noetherian means that every ideal is finitely generated; the dimensions and valuation overrings are as defined in this development.

External input

This is the classical noetherian-domain comparison recorded in the cited introduction. It is used as an external mathematical input; no local reconstruction of its proof is claimed.

Depends on

Used by · 1

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Parviz Sahandi, Semistar-Krull and Valuative Dimension of Integral Domains · arXiv:0809.1305v3, page 1, Section 1, introductory paragraph on classical valuative dimension

For a noetherian integral domain D, its Krull dimension equals the supremum of the Krull dimensions of its valuation overrings in Frac(D).

Hypotheses: D is a commutative noetherian integral domain with identity; dimension may be infinite.

Reading limits: Read the introductory classical statement and its definitions in the served PDF. The introduction attributes the result to Jaffard; the original proof and the later semistar arguments were not checked.

Sources