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.[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.

Proof

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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

  • for each nonzero ideal  , the module   is Noetherian.

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

  • each nonzero submodule   is such that   is not faithful.

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.  

References

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  1. ^ Matsumura 1989, Theorem 3.7. (i)
  2. ^ Matsumura 1989, A remark after Theorem 3.7.
  3. ^ Matsumura 1989, Theorem 3.6.
  4. ^ Matsumura 1989, Theorem 3.5.
  • Eakin, Paul M. Jr. (1968), "The converse to a well known theorem on Noetherian rings", Mathematische Annalen, 177 (4): 278–282, doi:10.1007/bf01350720, MR 0225767, S2CID 121169172
  • Nagata, Masayoshi (1968), "A type of subrings of a noetherian ring", Journal of Mathematics of Kyoto University, 8 (3): 465–467, doi:10.1215/kjm/1250524062, MR 0236162
  • Eisenbud, David (1970), "Subrings of Artinian and Noetherian rings", Mathematische Annalen, 185 (3): 247–249, doi:10.1007/bf01350264, MR 0262275, S2CID 15821722
  • Formanek, Edward; Jategaonkar, Arun Vinayak (1974), "Subrings of Noetherian rings", Proceedings of the American Mathematical Society, 46 (2): 181, doi:10.1090/s0002-9939-1974-0414625-5, MR 0414625
  • Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8 (2nd ed.), Cambridge: Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461

Further reading

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