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. [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.
Theorem — [3] Let be a commutative ring and a faithful finitely generated module over it. If the ascending chain condition holds on the submodules of the form for ideals , then 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. [4] 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.
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. [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.
Theorem — [3] Let be a commutative ring and a faithful finitely generated module over it. If the ascending chain condition holds on the submodules of the form for ideals , then 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. [4] 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.