Noetherian ring – Wikipedia

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Emmy Noether formalizes the properties of a particular family of rings now called Noetherian rings .

In mathematics, a Noetherian ring is a special case of ring, that is to say a set with an addition and multiplication compatible with the addition, in the sense of distribution.

Many mathematical questions are expressed in a context of ring, the endomorphisms of a vector space or a module on a ring, the algebraic whole of the algebraic theory of numbers, or the surfaces of algebraic geometry. If the rings are numerous, there are few properties common to the simplest examples such as relative integers or coefficient polynomials in a body. The Euclidean division generally no longer exists, the ideals, major tools of the theory of the rings, are no longer always main and the fundamental theorem of arithmetic no longer has an equivalent.

The approach consisting in studying a question only from the angle of the specific properties of a particular ring structure has proven to be fruitful. Richard Dedekind used it successfully in arithmetic and David Hilbert in algebraic geometry. In 1920-1921, Emmy Noether chose a more limited number of properties verified by certain rings and demonstrated many results on them.

The term “noetherian ring” appeared in 1943 under the pen of Claude Chevalley [ first ] .

In a main ring, all ideals are main. In other words, if the ring is considered to be a module on itself, its ideals are then submodules caused by an element. But many usual rings are not main. The ring ℤ [ X ] Polynomials with whole coefficients is an example of a non -main ring.

In arithmetic, it is common to use rings of algebraic whole, such as the ring ℤ [ i √ 5 ], which is an example of a ring of non -factory quadratic whole [ 2 ] So no principal. However, in ℤ [ i √ 5 ], all ideals are generated by one or two elements. More generally, in any ring of algebraic whole of a number body, the ideals, if not generated by a single element, are by a finished number of elements. A ring checking this property will be said to be Noetheran.

This configuration is found in group theory. If an Abélien group (seen as ℤ-model) is of the finite type (that is to say admits a finite generator part), all its subgroups are finished type sub-modules. The property is the same, even if it applies to a module and no longer to a ring. More generally, a finished type module in which any submodle is of the finished type is said to be Noetheran. This property is a good substitute for the hypothesis of the finished dimension in linear algebra.

Ring and module [ modifier | Modifier and code ]

Just as a commutative body is a vector space on itself, it is possible to consider a ring A like a A -module. If the ring is not commutative, there are two different external products. Be λ an element of A seen as a scalar and a an element of A seen as a vector, the two external products associate respectively with (λ, a ) Vectors λ. a And a .you. The anneau A thus has two structures of A -Modula, one on the left and the other on the right, which coincide if A is commutative.

A second difference lies in vector subspaces. A body contains only two: zero space and the body itself. For a ring A , considered as A -Module on the left (resp. On the right), the notion of sub-model coincides with that of ideal on the left (resp. On the right).

A ring A being always supposed to be unit in this article , the A -module A has a generating family made up of a single element: the unity (or any invertible element).

Noetherianity [ modifier | Modifier and code ]

Noetherianity is also simply defined on a module. The definition of Noetherian ring then becomes a special case, that where the ring is considered to be a module on itself (left or right).

  • And A – module M is said noetime If any submodle of M is finished
  • L’ ring A is said
    • Noetherian on the left If all its ideals on the left are of the finished type;
    • Noetherian on the right If all the ideals on the right are of the finished type;
    • noetime If he is a noetherian on the right and on the left.

In the case of commutative rings, these three definitions coincide [ 3 ] .

Either P a sub-model of M , the module M is noethérien if and only if P And M / P are.

Demonstration

  • If the module M is noethérien, any submodule P is too:
    Indeed, any submodle of P is a submodule of M So is of the finished type.
  • If the module M is noetherian, any quotient module M / P is too:
    Be Q a submodule of the quotient M / P , And R The reciprocal image of Q by the canonical projection of M In M / P . SO R is a submodule of M So has a finished generating family. The image of this family by the canonical projection generates Q , which shows that the quotient is also a noetherian.
  • If the submodule P and the quotient module M / P are noetheran then the module M is too:
    Be R a sub-model of M , F A finished family causing R ∩ P And G a finished family causing the image of R by the canonical projection of M In M / P . Let’s choose a finished family H vectors R whose projected is the elements of G . SO R is generated by the finished family F ∪ H .

We deduce immediately:

  • Any finished product of noetheran modules on the left is noetheran on the left;
  • Any quotient ring (by an ideal bilateral) of a noetherian ring on the left is noetheran on the left;
  • Any finished type module on a noetherian ring is noethérien [ 4 ] , [ 5 ] .

There are four alternative and equivalent definitions of the concept of Noetherian module (which immediately translate for the rings) [ 6 ] :

Either M and A -module. The following three properties are equivalent:

first. M is noethérien;

2. Any growing subposses of sub-models M is stationary;

2 bis. Any growing suite of finished sub-models of M is stationary;

3. Any non-empty set of sub-models of M admits a maximum element for inclusion;

3 bis. any non-empty set of finished sub-models of M admits a maximum element for inclusion.

Demonstration

Either ( M n ) an increasing suite of sub-models M . Note N The union of all modules M n . As they are nested, N is a submodule. By hypothesis 1, N admits a finished generator family ( m j ). Each of these vectors m j belongs to N , so to one of the M n (and to all the following): there is therefore an index n j such as this vector m j belongs to all M i For i greater than or equal to n j . Either n The maximum of the finished family of clues n j , and i is taller than n SO M i contains the family ( m j ) and so N . This shows that the suite of sub-models is constant from the rank n , therefore stationary.

We reason in contraded, we therefore assume that a non -empty set F sub-models does not have a maximum element. We will define a suite ( M n ) strictly growing elements of F . Either M 0 any element of F . We assume the suite defined to order n , M n is not a maximum element of F , we can therefore choose in F an element M n +1 strictly M n . There is then a strictly growing suite for inclusion. In contrap, the proposal is demonstrated.

Same reasoning, by replacing “sub-models” with “finished type sub-modulas”.

Property 3 bis being hereditary ( i.e. checked by any submodle of M As soon as it is verified by M ), just show that any module M Who checks it is finished type. Consider the whole F finished submodles of M . By hypothesis 3, F admits a maximum element N . Let us show M = N . Par construction, N is of the finished type: there is a finished family ( f i ) that generates it. Either m any element of M , consider the submodule P engendered by the family made up of m and f i : P is of the finished type so belongs to F . As N is maximum and that P contains it, N is equal to P . Consequently N Contains the vector m . As m is any vector of M , N is equal to M , which shows that the module M is of the finished type.

Properties 2 and 3 constitute the ascending chain condition on the sub-models of M . They make it possible to demonstrate the Krull theorem in the Noetherian case by a form of the axiom of the choice less elaborated than for a demonstration in the general case of a commutative ring: according to ownership 3 above,

Any clean ideal is included in a maximum ideal.

The decomposition of ideals is more delicate. In the main commutative ring ℤ For example, the ideal 12ℤ is equal both to the product of the ideals 2ℤ, 2ℤ and 3ℤ, and the intersection of ideals 2 2 ℤ and 3ℤ (which is also their product). In a commutative ring only a noetherian, three properties come close to it (the first is used in the article “Discreet valuation ring”, the fourth is the theorem of Lasker-Noether):

Either A A noetherian commutative ring.

  1. Any ideal for A contains a product of first ideals, or more precisely, any ideal I of A contains a product of first ideals which contain I [ 7 ] .
  2. For all ideal of A , there are a finite number of minimum first ideals containing this ideal.
  3. Any radical ideal of A is finished intersection of first ideals.
  4. Any ideal for A is decomposable, that is to say the finite intersection of primary ideals.
  5. And A is also honest so it is atomic, that is to say that any element of A non -zero and not invertible is produced by a finite number of irreducible elements [ 8 ] .

Demonstrations

  • Any ideal for A contains a product of first ideals, or more precisely, any ideal I of A contains a product of first ideals which contain I :
    Let us show the first assertion, by the absurd. Either F the whole, supposedly unused, of the ideals of A which contain no product of first ideals (especially no product indexed by a singleton, so the elements of F are not primary, and no product indexed by vacuum, so A does not belong to F ). Either I a maximum element of F . Ideal I is clean and not first. According to a property characteristic of non -first ideals, there are therefore two ideals J And K such as I contain the product J.K but be strictly contained in J And K . Then (by maximum of I ) the ideals J And K do not belong to F , so each of them contains a product of first ideals. As I Contains their product, we lead to a contradiction, which ends the demonstration of the first point.
    Due to the biunivia correspondence between the primary ideals of the quotient ring A/I and those of the ring A container I , we deduce the second point from the first applied to the ideal (0) of the ring A/I .
    A more complicated, but instructive proof, is to use the primary decomposition (stated below), noting that the radical of any primary ideal is first, and that in a noetherian ring, any ideal contains a power of its radical.
  • Existence and finitude of minimum first ideals containing an ideal I :
    We know that there are first ideals
    P1,…,Pn{displaystyle P_{1},dots ,P_{n}}

    whose product is included in I . An ideal first any Q of A container I contains in particular the product of Pi{displaystyle P_{i}}

    . A property characteristic of the primary ideals indicates that the first ideal Q contains one of Pi{displaystyle P_{i}}

    . Thus, the minimum elements among the ideals Pi{displaystyle P_{i}}

    are the minimum ideals containing I .
  • Any radical ideal of A is finished intersection of first ideals:
    Either I a radical ideal of A . We know that there are first ideals
    P1,…,Pn{displaystyle P_{1},dots ,P_{n}}

    such as ∏i=1nPi⊂I⊂⋂i=1nPi{displaystyle prod _{i=1}^{n}P_{i}subset Isubset bigcap _{i=1}^{n}P_{i}}

    . More x∈⋂i=1nPi{Displaystyle please BigCAP _ {i = 1}^{n} p_ {i}}

    SO xn∈∏i=1nPi⊂I{displaystyle x^{n}in prod _{i=1}^{n}P_{i}subset I}

    , SO x∈I=I{Displaystyle please {sqrt {i}} = i}

    (development), hence the equality I=⋂i=1nPi{Displaystyle i = BigCAP _ {i = 1}^{n} p_ {i}}

    .
  • Any ideal is finished intersection of primary ideals:
    We actually show a little better: in a noetherian ring, any ideal is finished intersection of irreducible ideals, and all irreducible ideal is primary.
    • For the first point, we reason by the absurd as before: be F the whole, supposedly unused, of the ideals of A which are not finite intersection of irreducible, and I a maximum element of F . SO I is therefore reducible equal to the intersection of two ideals J And K in which it is strictly included. By maximum, J And K do not belong to F , so each of them is an endless intersection of irreducible, hence the contradiction.
    • For the second point, cf. The algebraic challenge , volume 2, by Claude Mutafian, p. 239, or Bourbaki AC, chapter II, § 2, exercise 22, or (in) Oscar Zariski and Pierre Samuel, Commutative algebra , vol. first, chap.  IV, p. 209 .
      (We can also consult the primary decomposition article, which extends this theorem to finished type modules on a Noetherian ring, and most of the reference works like Lang 1965, or Bourbaki AC Chapter IV, which generalize this to the Noetheran modules on a any commutative ring.)
  • And A is honest then it is atomic:
    Indeed, for an integrated ring to be atomic, it is enough that it checks “ACCP” (the condition of ascending chain on the main ideals).

Any overjective endomorphism of a noetherian module is a self -corphism [ 9 ] .

First cases [ modifier | Modifier and code ]

Any commutative body is manifestly noetherian, by the absence of non -trivial ideals. Any main ring is also noetherian because each ideal is generated by a single element, so ℤ, K [ X ] The ring of coefficient polynomials in a body is noetheran. On the other hand, when possible, it is easier to study them using an Euclidean division or, which is always possible, to use the fundamental theorem of arithmetic as part of a ring factorial.

Any finished ring is noetherian, we find their presence, for example within the framework of algebraic geometry or the algebraic theory of numbers.

Polynomials and formal series [ modifier | Modifier and code ]

If a commutative ring A [( X i ) i ∈ I ] Polynomials in any number of indeterminate is noethérien, then the ring of coefficients A is clearly Noetheran. A reciprocal partial , discovered by David Hilbert in 1888 [ ten ] is named theorem of the Hilbert base:

Either A A noetherian commutative ring, the polynomial ring A [ X ] is noethérien.

It is easily generalized (by recurrence) in the case of a number fine indeterminate:

Be A a noetherian commutative ring and n A natural integer, the polynomial ring A [ X first , …, X n ] is noethérien.

On the other hand, a ring of polynomials on an infinite number of indeterminate is never a noetherian (whatever the ring of coefficients): the continuation of ideals whose n -th is engendered by ( X first , …, X n ) is growing but not stationary.

As an example of use, one can imagine in geometry an algebraic surface S Defined as all the roots of an infinite family of polynomials with several indeterminate and on a noetherian ring. The theorem of the Hilbert base indicates that it is enough to consider a finished family of polynomials to define S . Indeed, all the polynomials cancel out on S form an ideal.

By a similar argument (relating to the lower degree non -zero coefficients instead of the dominant coefficients), we demonstrate the following theorem (which is also generalized to several indeterminate) [ 11 ] :

Either A A noetherian commutative ring, the ring of formal series A [[ X ]] is noethérien.

Ring [ modifier | Modifier and code ]

Several examples of Noetherian rings come from arithmetic via the study of Diophantian equations, even if their use is now largely exceeding this framework. A simple example is given by the theorem of the two squares of Fermat, which involves the ring of the whole of Gauss. It is the ring of the integers of a quadratic body, therefore, like the ring of the whole of any body of numbers, it is a ring of dedekind and a finished type ℤ-module. In particular, he is a noetherian. More generally [ twelfth ] :

Be

A an integrated commutative ring,
K his body of fractions,
L a separable finite extension of K , And
B The ring of elements of L whole on A .

And A is noetherian and fully closed then B is a A -finished type module.

(The “whole element” article shows that B is a ring. Clearly, it contains A And it is commutative unit and honest.) Note that according to this statement, B is noetheran as a module, but also as a year , since it is a quotient of a ring of polynomials in a finite number of indeterminate to coefficients in A.

The demonstration proposed here uses the trace form, which also serves to define the discriminant of a ring. The form trace of L on K is not degenerate therefore the determinant Δ of its matrix M , in a base ( b first , …, b n ) of K -vector space L , is not zero. By choosing the b k In B , M is also to coefficients in A . It is enough, to conclude, to verify that B is included in the sub A – L generated by b first /D,…, b n /Δ, using the Laplace formula:

∀ b = ∑ xibi∈ B D (x1⋮xn)= (tcomM ) M (x1⋮xn)= tcomM(trL/K(b1b)⋮trL/K(bnb))∈ An. {displaystyle forall b=sum x_{i}b_{i}in Bquad Delta {begin{pmatrix}x_{1}\vdots \x_{n}end{pmatrix}}=(^{operatorname {t} }{rm {com}}M)M{begin{pmatrix}x_{1}\vdots \x_{n}end{pmatrix}}={^{operatorname {t} }{rm {com}}M}{begin{pmatrix}{rm {tr}}_{L/K}(b_{1}b)\vdots \{rm {tr}}_{L/K}(b_{n}b)end{pmatrix}}in A^{n}.}

In the same register, we also have:

Théorème de Krull-Akizuki [ 14 ] – Be A A commutative ring integrates noethérien whose first non -zero first ideal is maximum, K his body of fractions, L a finite extension of K , And B A subanary of L container A . SO B is noethérien, and any ideal first non -zero of B is maximum. In addition, for any non -zero ideal J of B , the A -module B/J is of the finished type.

Most algebraic operations retain noetherianity. Let us recall and complete the above examples:

  • The main bodies and rings are Noetherian;
  • Any quotient and finished direct product of Noetherian rings is Noetherian;
  • Any ring of polynomials has a finite number of indeterminate on a noetherian ring is noetheran. Thus any finished type algebra on a noetherian ring is noetherian;
  • Any located in a noetherian ring is noetheran; more generally, if M is a A -Noethérien module, all localized S −1 M is a S −1 A -Segle noethérien. (Indeed, for any submodle N of S −1 M , on a N = ( S −1 A ) ( N ∩ M ); We deduce that any growing follow -up ( N n ) sub-models of S −1 M is stationary, since the rest ( N n ∩ M ) is.)
  • The formal complement (in) of a noetherian commutative ring for the topological I -Adique ( I an ideal of A ) is noethérien;
  • If a noetherian ring is finished on a subanary (that is, it is finished type as a module on the subanary), then the latter is Noetheran (Eakin-Nagata theorem (in) );
  • Any ring is an increasing meeting of Noethian sub-year.

On the other hand, in general,

  • A subanary of a noetherian ring is not a noetherian (for example the ring of polynomials in infinity of indeterminates with coefficients in a body is not noetheran, but it is a subanary of its body fractions which is noetherian);
  • A Tensoriel Product of Noethian Rings is not Noetheran (take L the body of rational fractions to infinity of indeterminate to coefficients in a body K and consider the tensorial product
    L ⊗ KL {displaystyle Lotimes _{K}L}

    . The latter is not Noetheran, while K And L are).
  1. ↑ (in) Claude Chevalley, « On the Theory of Local Rings », The Annals of Mathematics , Second Series, Vol. 44, No. 4 (Oct., 1943), pp. 690-708.
  2. ↑ Because in this ring, 2 is irreducible but not first.
  3. ↑ It is even enough then – Cohen theorem – that all the ideals first of the ring are of the finished type: cf. N. Bourbaki, Commutative algebra , chap.  II , § 1, exercise 6 or (in) Joseph J. Rotman  (in) , Advanced Modern Algebra , AMS, ( read online ) , p. 320 , or (in) Irving Kaplansky, Commutative Rings , University of Chicago Press, ( first re ed. 1970), p. 5 .
  4. ↑ (in) Serge Lang, Algebra [Detail of editions] , 1965, p. 144 .
  5. ↑ (in) Michael art , Algebra [Edition detail] , p. 469 .
  6. ↑ The equivalence between 1, 2 and 3 is demonstrated for example in (in) Keith Conrad, ‘ Noetherian modules » , on math.uconn.edu , Th. 1.7 (for modules) and Th. 3.2 (for switching rings).
  7. ↑ In these two statements, of course, the rehearsals of the same first ideal in the product are authorized (counter-exampled otherwise, for the second: the ideal of the multiples of 4, in the ring of whole).
  8. ↑ This factorization is generally not unique even to multiplication near invertible. Thus the Noetherian ring A is factorial if and only if its irreducible elements are primary.
  9. ↑ (in) Alberto Facchini, Module Theory : Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules , Birkhäuser, coll.  « Progress in Mathematics » ( n O 167), , 288 p. (ISBN  978-3-7643-5908-9 , Online presentation ) , p. forty six .
  10. ↑ The proof of Hilbert caused a vast controversy in his time. The proof is indeed not constructive. Gordan, specialist in the question, exclaimed: It is not mathematics, it is theology , he ended up a few years later by admitting this proof and indicated: I have acquired the conviction that theology also has its advantages (J. Boniface, Hilbert and the notion of existence in mathematics , Philosophical Bookstore Vrin, 2004, chap. 2 p. 53 and chap. 1 p. 15 (ISBN  2711616061 ) ).
  11. ↑ Only 1965, p. 146-147.
  12. ↑ For more general results, cf. Bourbaki AC IX § 4.
  13. ↑ (in) David Eisenbud, Commutative Algebra : With a View Toward Algebraic Geometry , Springer, coll.  « GTM » ( n O 150), , 785 p. (ISBN  978-0-387-94269-8 , Online presentation ) , p. 298 .
  14. ↑ Bourbaki Ac VII, § 2, N ° 5, insight on Google Books .

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