In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, the Eakin–Nagata theorem states: given
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
s
such that
is
finitely generated as a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
over
, if
is a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
, then
is a Noetherian ring. (Note the
converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical c ...
is also true and is easier.)
The theorem is similar to the
Artin–Tate lemma In algebra, the Artin–Tate lemma, named after Emil Artin and John Tate, states:
:Let ''A'' be a commutative Noetherian ring and B \sub C commutative algebras over ''A''. If ''C'' is of finite type over ''A'' and if ''C'' is finite over ''B'', t ...
, which says that the same statement holds with "Noetherian" replaced by "
finitely generated algebra In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra ''A'' over a field ''K'' where there exists a finite set of elements ''a''1,...,''a'n'' of ''A'' such that every element of ...
" (assuming the base
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
is a Noetherian ring).
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
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and Director of the Mathematical Sciences Research Institute (MSRI); he previously serve ...
in ; this approach is useful for a generalization to
non-commutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not a ...
s.
Proof
The following more general result is due to
Edward W. Formanek
Edward William Formanek (born May 6, 1942). is an American mathematician and chess player. He is a professor emeritus of mathematics at Pennsylvania State University,.. and a International Master, FIDE International Master in chess.[Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.
Historically, Hilbert was the first mathematician to work with the propert ...]
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
*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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
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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*
*
*
*
Further reading
Math StackExchange - Exercise from Kaplansky's Commutative Rings and Eakin-Nagata Theorem
{{DEFAULTSORT:Eakin-Nagata theorem
Theorems in ring theory
Commutative algebra