Algebra · existing Mathlib declaration

Hilbert Basis Theorem

If every ideal in a commutative ring can be generated by finitely many elements, the same remains true after adjoining one polynomial variable.

Hilbert basis theorem
Noetherian rings
univariate polynomial rings
finitely generated ideals
leading coefficients
degree induction

The theorem at a glance

Hilbert Basis Theorem — finite generation survives one polynomial variable

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

An ivory commutative-algebra poster states that a Noetherian commutative ring has a Noetherian univariate polynomial ring, then shows leading-coefficient ideals stabilizing and higher-degree terms descending into a finite span. — For a commutative Noetherian ring R, every ideal of the one-variable polynomial ring R[X] is finitely generated.

Formal orientation

Statement map

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

Passed full-canvas review and lossless 1:1 quadrant inspection. The distinct Scientific diagram gives a precise left-to-right dependency route from R NOETHERIAN through the leading-coefficient chain and its stable plateau, a visibly finite bounded-degree generator set, leading-term cancellation with degree descent, I = span(s), and the exact R[X] NOETHERIAN endpoint. Every required label is complete and legible, the one-variable structure is explicit, the targeted correction preserves ⊆ and removes the stray subtraction glyph, and there is no multivariable, field-only, algorithmic, or novelty overclaim. — Noetherian control in the coefficient ring stabilizes leading coefficients and lets finitely many bounded-degree polynomials generate an arbitrary ideal of R[X].

Complementary intuition

A mathematical landmark

Hilbert's basis theorem is a foundational finiteness result in commutative algebra: adjoining one polynomial variable preserves the Noetherian condition and keeps every polynomial ideal finitely generated.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

protected theorem Polynomial.isNoetherianRing {R : Type u} [CommRing R] [inst : IsNoetherianRing R] : IsNoetherianRing R[X]

Selected evidence declaration text

protected theorem Polynomial.isNoetherianRing {R : Type u} [CommRing R] [inst : IsNoetherianRing R] : IsNoetherianRing R[X]

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's protected theorem that a commutative Noetherian ring R has a Noetherian univariate polynomial ring R[X]. The conclusion is the IsNoetherianRing R[X] typeclass proposition, and the source registers the theorem as an instance on the following line. It is not a multivariate endpoint, a generator-computation algorithm, or new ProofAtlas mathematics.

The exact declaration adjoins one polynomial variable; it does not itself state a multivariable polynomial theorem.
The theorem produces a typeclass proposition, not an explicit algorithm for computing generators of an ideal.
The source assumes a commutative ring; this page does not extend the result to arbitrary semirings or noncommutative polynomial constructions.
ProofAtlas is indexing an existing Mathlib theorem, not claiming mathematical novelty.

Provenance

Origin and evidence stay separate

Existing declaration: Polynomial.isNoetherianRing in mathlib
Relationship: The checked artifact is the upstream declaration itself
Preferred checked artifact: artifact.library.mathlib.hilbert-basis-theorem.v001
Source handling: Verified upstream reference; no mirrored source package