Commutative Algebra pp 443-465 | Cite as

# A Method for the Study of Artinian Modules, With an Application to Asymptotic Behavior

## Abstract

If *N* is a Noetherian module over the commutative ring *R* (throughout the paper, *R* will denote a commutative ring with identity), then the study of *N* in many contexts can be reduced to the study of a finitely generated module over a commutative Noetherian ring, because *N* has a natural structure as a module over *R*/(0 : *N*) and the latter ring is Noetherian. For a long time it has been a source of irritation to me that I did not know of any method which would reduce the study of an Artinian module *A* over the commutative ring *R* to the study of an Artinian module over a commutative Noetherian ring. However, during the MSRI Microprogram on Commutative Algebra, my attention was drawn to a result of W. Heinzer and D. Lantz [**2**, Proposition 4.3]; this proposition proves that if *A* is a faithful Artinian module over a quasi-local ring (*R*, *M*) which is (Hausdorff) complete in the *M*-adic topology, then *R* is Noetherian. It turns out that a generalization of this result provides a missing link to complete a chain of reductions by which one can, for some purposes, reduce the study of an Artinian module over an arbitrary commutative ring *R* to the study of an Artinian module over a complete (Noetherian) local ring; in the latter situation we have Matlis’s duality available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

## Keywords

Local Ring Maximal Ideal Commutative Ring Homomorphic Image Noetherian Ring## Preview

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