Eakin–Nagata theorem

In abstract algebra, the Eakin–Nagata theorem states: given commutative rings

is finitely generated as a module over

is a Noetherian ring, then

is a Noetherian ring.

[1] (Note the converse is also true and is easier.)

The theorem is similar to the Artin–Tate lemma, which says that the same statement holds with "Noetherian" replaced by "finitely generated algebra" (assuming the base ring is a Noetherian ring).

The theorem was first proved in Paul M. Eakin's thesis (Eakin 1968) and later independently by Masayoshi Nagata (1968).

[2] 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 (Eisenbud 1970); this approach is useful for a generalization to non-commutative rings.

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 (Matsumura 1989), this formulation is likely the most transparent one.

be a commutative ring and

a faithful finitely generated module over it.

If the ascending chain condition holds on the submodules of the form

is a Noetherian ring.

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.

By assumption, the set of all

is not Noetherian has a maximal element,

, we can assume Next, consider the set

Choose a set of generators

Thus, it is clear that Zorn's lemma applies to the set

, and so the set has a maximal element,

is Noetherian, then it is a faithful Noetherian module over A and, consequently, A is a Noetherian ring, a contradiction.

is not Noetherian and replacing

, we can also assume Let a submodule

is not faithful, there is a nonzero element

is finitely generated.

is also finitely generated, it follows that

is finitely generated; i.e.,

is Noetherian, a contradiction.