The Artin-Rees Lemma states that for a Noetherian ring, an ideal, a finitely generated module over that ring, and a submodule of it, the descending chain formed by intersecting the submodule with successive powers of the ideal times the whole module eventually behaves the same as the chain generated by the ideal acting on the submodule itself, up to a fixed shift in index. Named for Emil Artin and David Rees, it is a technical but foundational tool of commutative algebra underlying results such as the Krull intersection theorem and the construction of formal completions.
Facts
StatementFor a Noetherian ring R with ideal I, a finitely generated R-module M and a submodule N of M, there is an integer k of at least 1 such that for every n of at least k, the intersection of I to the n times M with N equals I to the n minus k times the intersection of I to the k times M with N. 1 Proof YearRees's paper, cited here, was published in 1956; the article's own opening sentence says Emil Artin reached the same result independently around the same time but gives no separate publication year for Artin's side. Classification
Statement Form Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Proved By
Source Artin-Rees Lemma (Wikipedia)
Sources
1. Artin-Rees Lemma (Wikipedia)
Wikimedia FoundationStatement section, the lemma itself
Let I be an ideal in a Noetherian ring R; let M be a finitely generated R-module and let N a submodule of M. Then there exists an integer k ≥ 1 so that, for n ≥ k,
References list, Rees's 1956 paper
Rees, David (1956). Two classical theorems of ideal theory. Mathematical Proceedings of the Cambridge Philosophical Society. 52 (1): 155-157.
Proved By: Emil Artin, Lead paragraph
In mathematics, the Artin-Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem.
View the Source Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.