{ "artifactId": "artifact.library.mathlib.hilbert-basis-theorem.v001", "candidateOnly": false, "category": "Algebra", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "protected theorem Polynomial.isNoetherianRing {R : Type u} [CommRing R] [inst : IsNoetherianRing R] : IsNoetherianRing R[X]", "family": "Commutative algebra", "id": "library.mathlib.hilbert-basis-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/hilbert-basis-theorem/", "entryData": "/data/library-theorems/hilbert-basis-theorem.json", "evidence": "/proofs/artifact.library.mathlib.hilbert-basis-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.hilbert-basis-theorem.v001.evidence.json", "source": "/sources/upstream/hilbert-basis-theorem/" }, "landmark": { "completeFormalType": "protected theorem Polynomial.isNoetherianRing {R : Type u} [CommRing R] [inst : IsNoetherianRing R] : IsNoetherianRing R[X]", "concepts": [ "Hilbert basis theorem", "Noetherian rings", "univariate polynomial rings", "finitely generated ideals", "leading coefficients", "degree induction" ], "plainLanguage": "If every ideal in a commutative ring can be generated by finitely many elements, the same remains true after adjoining one polynomial variable.", "poster": { "alt": "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.", "byteSize": 2961770, "caption": "For a commutative Noetherian ring R, every ideal of the one-variable polynomial ring R[X] is finitely generated.", "description": "The poster centers the exact transfer R Noetherian to R[X] Noetherian. Its engraved proof route tracks leading coefficients by degree, marks a stabilization threshold, finitely generates the bounded-degree part, and cancels higher leading terms to induct downward in degree.", "derivatives": [ { "byteSize": 117372, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hilbert-basis-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:983f86314a7999b7a6ffe7159bb67907d33466bf1f4bd180953c0aaa7f1e40de", "width": 640 }, { "byteSize": 280578, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hilbert-basis-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:468ad81104c1ec5b9bb3e3d742486e21547a132625459cc49846515884f03ae8", "width": 1024 } ], "generatedAt": "2026-07-25T16:26:46-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/hilbert-basis-theorem/theorem-poster-v1.png", "sha256": "sha256:2c61bd1caccf63d5b827894321670bdd47969e153537f845eba5b8393312f8f9", "title": "Hilbert Basis Theorem — finite generation survives one polynomial variable", "width": 1024, "kind": "theorem_poster", "transcript": "COMMUTATIVE ALGEBRA · FINITE GENERATION\nHILBERT BASIS THEOREM\nR a commutative ring · R Noetherian\nR[X] is Noetherian\nONE VARIABLE, EVERY IDEAL FINITELY GENERATED\nHOW THE CHECKED ROUTE MOVES\n1 · Track an ideal's leading coefficients by degree\n2 · Choose N where the ascending family stabilizes\n3 · Finitely generate the part of degree at most N\n4 · Cancel a higher leading term and induct on degree\nEXACT SCOPE\nUnivariate protected theorem. No multivariate or novelty claim." }, "visual": { "alt": "A stable ring lattice supports one ascending polynomial tower whose many strands are gathered by three gold clasps, while a side chain of leading-coefficient medallions reaches a fixed plateau.", "byteSize": 3018575, "caption": "Noetherian control in the coefficient ring stabilizes leading coefficients and lets finitely many bounded-degree polynomials generate an arbitrary ideal of R[X].", "description": "One engraved polynomial direction rises from a compact coefficient-ring lattice. A finite cluster of clasps spans the ideal's many strands, while a monotone side chain reaches a gold threshold and higher strands descend by leading-term cancellation.", "derivatives": [ { "byteSize": 47734, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hilbert-basis-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:7155199ddc19110df95ff7a4f485f5b4e7ef60377356093133168104d5b11076", "width": 640 }, { "byteSize": 162394, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hilbert-basis-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:05466664d39f6d3ecc4c6ff88c29353ca46aa04dd2f01e88a7de0c3da9d751cb", "width": 1200 } ], "generatedAt": "2026-07-25T16:26:46-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/hilbert-basis-theorem/theorem-schematic-v1.png", "sha256": "sha256:675eef4884bf2d3dd68e5348b62999d350fe8f42401abb6217b8e33db53b28a0", "title": "Finite generators gather the univariate polynomial tower", "width": 1536 }, "walkthrough": null, "whyLandmark": "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." }, "nonClaims": [ "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." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.hilbert-basis-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "A commutative Noetherian ring remains Noetherian after adjoining one polynomial variable.", "targetId": "target.library.mathlib.hilbert-basis-theorem.v001", "title": "Hilbert Basis Theorem", "upstreamOrigin": { "declarationName": "Polynomial.isNoetherianRing", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/RingTheory/Polynomial/Basic.lean#L733", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:7cedafd3e1fc910b152c699375e8670f0db7d6261d7ebdd3dd8ff2420fda5b9c", "sourceByteLength": 40333, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/RingTheory/Polynomial/Basic.lean", "sourceLine": 733, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }