Artin–Rees lemma

The Artin–Rees lemma states that if a submodule lies in a finitely generated module over a Noetherian ring, its induced ideal-adic filtration eventually agrees with the ambient filtration up to a fixed shift.

Rees algebra: The graded algebra formed from the powers of an ideal, encoding its filtration in a single algebraic object. Its Noetherianity supplies the finite-generation argument behind the lemma.
Finitely generated module: A module generated by finitely many elements over a ring. Finite generation is the module-side hypothesis that makes the filtration argument work.
Adic completion: The inverse limit of a module modulo successive powers of an ideal. Artin–Rees helps prove that completion preserves exact sequences of finitely generated modules.
Nakayama's lemma: A lemma stating that a finitely generated module over a local ring cannot equal its maximal-ideal multiple unless it is zero. It converts filtration information into generation or vanishing conclusions in local applications.
Rees module: A graded module formed from the powers of an ideal acting on a module. The graded submodule for the intersected filtration is finitely generated, forcing eventual stabilization.
Submodule: A subset of a module closed under addition and scalar multiplication. The theorem measures how an ambient ideal-adic filtration restricts to a submodule.
Exactness of completion: The property that completing a short exact sequence of modules preserves its exactness. The lemma supplies the filtration comparison needed for this fundamental consequence.
Noetherian module: A module in which every submodule is finitely generated, equivalently every ascending chain of submodules stabilizes. The lemma can be framed using module-theoretic Noetherianity rather than a Noetherian base ring.
Ideal-adic filtration: The descending sequence of submodules obtained by multiplying a module by successive powers of an ideal. The lemma compares this filtration on a submodule with the filtration inherited from its ambient module.
Ideal: An additive subgroup of a ring closed under multiplication by arbitrary ring elements. Its powers define the filtration whose induced behavior the theorem constrains.

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Jacobson's conjectureRelated: Its commutative filtration control helps explain why intersection arguments can succeed under stronger hypotheses.