Eakin–Nagata theorem


In abstract algebra, the Eakin–Nagata theorem states: given commutative rings such that is finitely generated as a module over, if is a Noetherian ring, then is a Noetherian ring.
The theorem is similar to the Artin–Tate lemma, which says that the same statement holds with "Noetherian" replaced by "finitely generated algebra".
The theorem was first proved in Paul M. Eakin's thesis and later independently by. The theorem can also be deduced from the characterization of a Noetherian ring in terms of injective modules, as done for example by David Eisenbud in ; this approach is useful for a generalization to non-commutative rings.

Proof

The following more general result is due to Edward W. Formanek and is proved by an argument rooted to the original proofs by Eakin and Nagata. According to, this formulation is likely the most transparent one.
Proof: It is enough to show that is a Noetherian module since, in general, a ring admitting a faithful Noetherian module over it is a Noetherian ring. Suppose otherwise. By assumption, the set of all, where is an ideal of such that is not Noetherian has a maximal element,. Replacing and by and, we can assume
Next, consider the set of submodules such that is faithful. Choose a set of generators of and then note that is faithful if and only if for each, the inclusion implies. Thus, it is clear that Zorn's lemma applies to the set, and so the set has a maximal element,. Now, if is Noetherian, then it is a faithful Noetherian module over A and, consequently, A is a Noetherian ring, a contradiction. Hence, is not Noetherian and replacing by, we can also assume
Let a submodule be given. Since is not faithful, there is a nonzero element such that. By assumption, is Noetherian and so is finitely generated. Since is also finitely generated, it follows that is finitely generated; i.e., is Noetherian, a contradiction.