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Noetherian ring
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{{short description|Mathematical ring with well-behaved ideals}} In [[mathematics]], a '''Noetherian ring''' is a [[ring (mathematics)|ring]] that satisfies the [[ascending chain condition]] on left and right [[Ideal (ring theory)|ideals]]; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said '''left-Noetherian''' or '''right-Noetherian''' respectively. That is, every increasing sequence <math>I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots</math> of left (or right) ideals has a largest element; that is, there exists an {{math|''n''}} such that: <math>I_{n}=I_{n+1}=\cdots.</math> Equivalently, a ring is left-Noetherian (respectively right-Noetherian) if every left ideal (respectively right-ideal) is [[finitely generated ideal|finitely generated]]. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both [[commutative ring|commutative]] and [[noncommutative ring|noncommutative]] ring theory since many rings that are encountered in mathematics are Noetherian (in particular the [[ring of integers]], [[polynomial ring]]s, and [[ring of algebraic integers|rings of algebraic integers]] in [[number field]]s), and many general theorems on rings rely heavily on the Noetherian property (for example, the [[Lasker–Noether theorem]] and the [[Krull intersection theorem]]). Noetherian rings are named after [[Emmy Noether]], but the importance of the concept was recognized earlier by [[David Hilbert]], with the proof of [[Hilbert's basis theorem]] (which asserts that polynomial rings are Noetherian) and [[Hilbert's syzygy theorem]]. {{Algebraic structures |Ring}} == Characterizations == For [[noncommutative ring]]s, it is necessary to distinguish between three very similar concepts: * A ring is '''left-Noetherian''' if it satisfies the ascending chain condition on left ideals. * A ring is '''right-Noetherian''' if it satisfies the ascending chain condition on right ideals. * A ring is '''Noetherian''' if it is both left- and right-Noetherian. For [[commutative ring]]s, all three concepts coincide, but in general they are different. There are rings that are left-Noetherian and not right-Noetherian, and vice versa. There are other, equivalent, definitions for a ring ''R'' to be left-Noetherian: * Every left ideal ''I'' in ''R'' is [[Ideal_(ring_theory)#Types_of_ideals|finitely generated]], i.e. there exist elements <math>a_1, \ldots , a_n</math> in ''I'' such that <math>I=Ra_1 + \cdots + Ra_n</math>.<ref name=":0">Lam (2001), p. 19</ref> * Every [[empty set|non-empty]] set of left ideals of ''R'', [[partially ordered]] by inclusion, has a [[maximal element]].<ref name=":0" /> Similar results hold for right-Noetherian rings. The following condition is also an equivalent condition for a ring ''R'' to be left-Noetherian and it is [[David Hilbert|Hilbert]]'s original formulation:<ref>{{harvnb|Eisenbud|1995|loc=Exercise 1.1.}}</ref> *Given a sequence <math>f_1, f_2, \dots</math> of elements in ''R'', there exists an integer <math>n</math> such that each <math>f_i</math> is a finite [[linear combination]] <math display="inline">f_i = \sum_{j=1}^n r_j f_j</math> with coefficients <math>r_j</math> in ''R''. For a commutative ring to be Noetherian it suffices that every [[prime ideal]] of the ring is finitely generated.<ref>{{Cite journal|last=Cohen|first=Irvin S.|author-link=Irvin Cohen|date=1950|title=Commutative rings with restricted minimum condition|url=https://projecteuclid.org/euclid.dmj/1077475897| journal=[[Duke Mathematical Journal]]|language=en|volume=17|issue=1|pages=27–42|doi=10.1215/S0012-7094-50-01704-2|issn=0012-7094|url-access=subscription}}</ref> However, it is not enough to ask that all the [[maximal ideal]]s are finitely generated, as there is a non-Noetherian [[local ring]] whose maximal ideal is [[principal ideal|principal]] (see a counterexample to Krull's intersection theorem at [[Local ring#Commutative case]].) == Properties == * If ''R'' is a Noetherian ring, then the [[polynomial ring]] <math>R[X]</math> is Noetherian by the [[Hilbert's basis theorem]]. By [[mathematical induction|induction]], <math>R[X_1, \ldots, X_n]</math> is a Noetherian ring. Also, {{math|''R''<nowiki>[[</nowiki>''X''<nowiki>]]</nowiki>}}, the [[Formal power series|power series ring]], is a Noetherian ring. * If {{math|''R''}} is a Noetherian ring and {{math|''I''}} is a two-sided ideal, then the [[quotient ring]] {{math|''R''/''I''}} is also Noetherian. Stated differently, the [[image (mathematics)|image]] of any [[surjective]] [[ring homomorphism]] of a Noetherian ring is Noetherian. * Every finitely-generated [[commutative algebra (structure)|commutative algebra]] over a commutative Noetherian ring is Noetherian. (This follows from the two previous properties.) * A ring ''R'' is left-Noetherian [[if and only if]] every finitely generated left [[module (mathematics)|''R''-module]] is a [[Noetherian module]]. * If a commutative ring admits a [[faithful module|faithful]] Noetherian module over it, then the ring is a Noetherian ring.<ref>{{harvnb|Matsumura|1989|loc=Theorem 3.5.}}</ref><!-- not sure if “commutative” can be dropped. --> * ([[Eakin–Nagata theorem|Eakin–Nagata]]) If a ring ''A'' is a [[subring]] of a commutative Noetherian ring ''B'' such that ''B'' is a [[finitely generated module]] over ''A'', then ''A'' is a Noetherian ring.<ref>{{harvnb|Matsumura|1989|loc=Theorem 3.6.}}</ref> *Similarly, if a ring ''A'' is a subring of a commutative Noetherian ring ''B'' such that ''B'' is [[faithfully flat ring homomorphism|faithfully flat]] over ''A'' (or more generally exhibits ''A'' as a [[pure subring]]), then ''A'' is a Noetherian ring (see the "faithfully flat" article for the reasoning). * Every [[Localization of a ring|localization]] of a commutative Noetherian ring is Noetherian. * A consequence of the [[Hopkins–Levitzki theorem|Akizuki–Hopkins–Levitzki theorem]] is that every left [[Artinian ring]] is left Noetherian. Another consequence is that a left Artinian ring is right Noetherian if and only if it is right Artinian. The analogous statements with "right" and "left" interchanged are also true. * A left Noetherian ring is left [[coherent ring|coherent]] and a left Noetherian [[Domain (ring theory)|domain]] is a left [[Ore domain]]. * (Bass) A ring is (left/right) Noetherian if and only if every [[direct sum of modules|direct sum]] of [[injective module|injective]] (left/right) modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct sum of [[indecomposable module|indecomposable]] injective modules.<ref name="Bass injective">{{harvnb|Anderson|Fuller|1992|loc=Proposition 18.13.}}</ref> See also [[#Implication on injective modules]] below. * In a commutative Noetherian ring, there are only finitely many [[minimal prime ideal]]s. Also, the [[descending chain condition]] holds on prime ideals. * In a commutative Noetherian domain ''R'', every element can be factorized into [[irreducible element]]s (in short, ''R'' is a [[factorization domain]]). Thus, if, in addition, the factorization is unique [[up to]] multiplication of the factors by [[unit (ring theory)|unit]]s, then ''R'' is a [[unique factorization domain]]. == Examples == * Any field, including the fields of [[rational number]]s, [[real number]]s, and [[complex number]]s, is Noetherian. (A field only has two ideals — itself and (0).) * Any [[principal ideal ring]], such as the integers, is Noetherian since every ideal is generated by a single element. This includes [[principal ideal domain]]s and [[Euclidean domain]]s. * A [[Dedekind domain]] (e.g., [[ring of integers|rings of integers]]) is a Noetherian domain in which every ideal is generated by at most two elements. * The [[coordinate ring]] of an [[affine variety]] is a Noetherian ring, as a consequence of the Hilbert basis theorem. * The enveloping algebra ''U'' of a finite-dimensional [[Lie algebra]] <math>\mathfrak{g}</math> is a both left and right Noetherian ring; this follows from the fact that the [[associated graded ring]] of ''U'' is a quotient of <math>\operatorname{Sym}(\mathfrak{g})</math>, which is a polynomial ring over a field (the [[PBW theorem]]); thus, Noetherian.<ref>{{harvnb|Bourbaki|1989|loc=Ch III, §2, no. 10, Remarks at the end of the number}}</ref> For the same reason, the [[Weyl algebra]], and more general rings of [[differential operator]]s, are Noetherian.<ref>{{harvtxt|Hotta|Takeuchi|Tanisaki|2008|loc=§D.1, Proposition 1.4.6}}</ref> * The ring of polynomials in finitely-many variables over the integers or a field is Noetherian. Rings that are not Noetherian tend to be (in some sense) very large. Here are some examples of non-Noetherian rings: * The ring of polynomials in infinitely-many variables, ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, etc. The sequence of ideals (''X''<sub>1</sub>), (''X''<sub>1</sub>, ''X''<sub>2</sub>), (''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>), etc. is ascending, and does not terminate. * The ring of all [[algebraic integer]]s is not Noetherian. For example, it contains the infinite ascending chain of principal ideals: (2), (2<sup>1/2</sup>), (2<sup>1/4</sup>), (2<sup>1/8</sup>), ... * The ring of [[continuous function]]s from the real numbers to the real numbers is not Noetherian: Let ''I<sub>n</sub>'' be the ideal of all continuous functions ''f'' such that ''f''(''x'') = 0 for all ''x'' ≥ ''n''. The sequence of ideals ''I''<sub>0</sub>, ''I''<sub>1</sub>, ''I''<sub>2</sub>, etc., is an ascending chain that does not terminate. * The ring of [[Homotopy groups of spheres#Ring structure|stable homotopy groups of spheres]] is not Noetherian.<ref>[https://math.stackexchange.com/q/1513353 The ring of stable homotopy groups of spheres is not noetherian]</ref> However, a non-Noetherian ring can be a subring of a Noetherian ring. Since any [[integral domain]] is a subring of a field, any integral domain that is not Noetherian provides an example. To give a less trivial example, * The ring of [[rational function]]s generated by ''x'' and ''y''/''x''<sup>''n''</sup> over a field ''k'' is a subring of the field ''k''(''x'',''y'') in only two variables. Indeed, there are rings that are right Noetherian, but not left Noetherian, so that one must be careful in measuring the "size" of a ring this way. For example, if ''L'' is a [[subgroup]] of '''Q'''<sup>2</sup> [[isomorphic]] to '''Z''', let ''R'' be the ring of homomorphisms ''f'' from '''Q'''<sup>2</sup> to itself satisfying ''f''(''L'') ⊂ ''L''. Choosing a basis, we can describe the same ring ''R'' as :<math>R=\left\{\left.\begin{bmatrix}a & \beta \\0 & \gamma \end{bmatrix} \, \right\vert\, a\in \mathbf{Z}, \beta\in \mathbf{Q},\gamma\in \mathbf{Q}\right\}.</math> This ring is right Noetherian, but not left Noetherian; the subset ''I'' ⊂ ''R'' consisting of elements with ''a'' = 0 and ''γ'' = 0 is a left ideal that is not finitely generated as a left ''R''-module. If ''R'' is a commutative subring of a left Noetherian ring ''S'', and ''S'' is finitely generated as a left ''R''-module, then ''R'' is Noetherian.<ref>{{harvnb|Formanek|Jategaonkar|1974|loc=Theorem 3}}</ref> (In the special case when ''S'' is commutative, this is known as [[Eakin–Nagata theorem|Eakin's theorem]].) However, this is not true if ''R'' is not commutative: the ring ''R'' of the previous paragraph is a subring of the left Noetherian ring ''S'' = Hom('''Q'''<sup>2</sup>, '''Q'''<sup>2</sup>), and ''S'' is finitely generated as a left ''R''-module, but ''R'' is not left Noetherian. A [[unique factorization domain]] is not necessarily a Noetherian ring. It does satisfy a weaker condition: the [[ascending chain condition on principal ideals]]. A ring of polynomials in infinitely-many variables is an example of a non-Noetherian unique factorization domain. A [[valuation ring]] is not Noetherian unless it is a principal ideal domain. It gives an example of a ring that arises naturally in [[algebraic geometry]] but is not Noetherian. === Noetherian group rings === Consider the [[group ring]] <math>R[G]</math> of a [[group (mathematics)|group]] <math>G</math> over a [[ring (mathematics)|ring]] <math>R</math>. It is a [[ring (mathematics)|ring]], and an [[associative algebra]] over <math>R</math> if <math>R</math> is [[commutative ring|commutative]]. For a group <math>G</math> and a commutative ring <math>R</math>, the following two conditions are equivalent. * The ring <math>R[G]</math> is left-Noetherian. * The ring <math>R[G]</math> is right-Noetherian. This is because there is a [[bijection]] between the left and right ideals of the group ring in this case, via the <math>R</math>-[[associative algebra]] [[homomorphism]] :<math>R[G]\to R[G]^{\operatorname{op}},</math> :<math>g\mapsto g^{-1}\qquad(\forall g\in G).</math> Let <math>G</math> be a group and <math>R</math> a ring. If <math>R[G]</math> is left/right/two-sided Noetherian, then <math>R</math> is left/right/two-sided Noetherian and <math>G</math> is a [[Noetherian group]]. Conversely, if <math>R</math> is a Noetherian commutative ring and <math>G</math> is an [[group extension|extension]] of a [[Noetherian group|Noetherian]] [[solvable group]] (i.e. a [[polycyclic group]]) by a [[finite group]], then <math>R[G]</math> is two-sided Noetherian. On the other hand, however, there is a [[Noetherian group]] <math>G</math> whose group ring over any Noetherian commutative ring is not two-sided Noetherian.<ref name="Ol’shanskiĭ">{{cite book |last1=Ol’shanskiĭ |first1=Aleksandr Yur’evich |title=Geometry of defining relations in groups |translator-last=Bakhturin |translator-first=Yu. A. |language=en |series=Mathematics and Its Applications. Soviet Series |volume=70 |publisher=Kluwer Academic Publishers |location=Dordrecht |date=1991 |isbn=978-0-7923-1394-6 |issn=0169-6378 |doi=10.1007/978-94-011-3618-1 |mr=1191619 |zbl=0732.20019 }}</ref>{{rp|423, Theorem 38.1}} == Key theorems == Many important theorems in ring theory (especially the theory of [[commutative ring]]s) rely on the assumptions that the rings are Noetherian. ===Commutative case=== *Over a commutative Noetherian ring, each ideal has a [[primary decomposition]], meaning that it can be written as an [[intersection (set theory)|intersection]] of finitely many [[primary ideal]]s (whose [[radical of an ideal|radical]]s are all distinct) where an ideal ''Q'' is called primary if it is [[proper ideal|proper]] and whenever ''xy'' ∈ ''Q'', either ''x'' ∈ ''Q'' or ''y''<sup> ''n''</sup> ∈ ''Q'' for some positive integer ''n''. For example, if an element <math>f = p_1^{n_1} \cdots p_r^{n_r}</math> is a product of powers of distinct prime elements, then <math>(f) = (p_1^{n_1}) \cap \cdots \cap (p_r^{n_r})</math> and thus the primary decomposition is a direct generalization of [[prime factorization]] of integers and polynomials.<ref>{{harvnb|Eisenbud|1995|loc=Proposition 3.11.}}</ref> *A Noetherian ring is defined in terms of ascending chains of ideals. The [[Artin–Rees lemma]], on the other hand, gives some information about a descending chain of ideals given by powers of ideals <math>I \supseteq I^2 \supseteq I^3 \supseteq \cdots </math>. It is a technical tool that is used to [[mathematical proof|prove]] other key theorems such as the [[Krull intersection theorem]]. *The [[dimension theory (algebra)|dimension theory]] of commutative rings behaves poorly over non-Noetherian rings; the very fundamental theorem, [[Krull's principal ideal theorem]], already relies on the "Noetherian" assumption. Here, in fact, the "Noetherian" assumption is often not enough and (Noetherian) [[universally catenary ring]]s, those satisfying a certain dimension-theoretic assumption, are often used instead. Noetherian rings appearing in applications are mostly universally catenary. ===Non-commutative case=== {{expand section|date=December 2019}} *[[Goldie's theorem]] == Implication on injective modules == Given a ring, there is a close connection between the behaviors of [[injective module]]s over the ring and whether the ring is a Noetherian ring or not. Namely, given a ring ''R'', the following are equivalent: *''R'' is a left Noetherian ring. *(Bass) Each direct sum of injective left ''R''-modules is injective.<ref name="Bass injective" /> *Each injective left ''R''-module is a direct sum of [[indecomposable module|indecomposable]] injective modules.<ref>{{harvnb|Anderson|Fuller|1992|loc=Theorem 25.6. (b)}}</ref> *(Faith–Walker) There exists a [[cardinal number]] <math>\mathfrak{c}</math> such that each injective left module over ''R'' is a direct sum of <math>\mathfrak{c}</math>-generated modules (a module is <math>\mathfrak{c}</math>-generated if it has a [[generating set of a module|generating set]] of [[cardinality]] at most <math>\mathfrak{c}</math>).<ref>{{harvnb|Anderson|Fuller|1992|loc=Theorem 25.8.}}</ref> *There exists a left ''R''-module ''H'' such that every left ''R''-module [[embedding|embeds]] into a direct sum of copies of ''H''.<ref>{{harvnb|Anderson|Fuller|1992|loc=Corollary 26.3.}}</ref> <!--Expand this later: Over a commutative ring, decomposing an injective module is essentially the same as doing a primary decomposition and that explains "Noetherian" assumption. --> The [[endomorphism ring]] of an indecomposable injective module is [[local ring|local]]<ref>{{harvnb|Anderson|Fuller|1992|loc=Lemma 25.4.}}</ref> and thus [[Azumaya's theorem]] says that, over a left Noetherian ring, each indecomposable decomposition of an injective module is equivalent to one another (a variant of the [[Krull–Schmidt theorem]]). == See also == *[[Noetherian scheme]] *[[Artinian ring]] ==Notes== {{reflist}} ==References== * {{citation |last1=Anderson |first1=Frank W. |last2=Fuller |first2=Kent R. |title=Rings and categories of modules |series=[[Graduate Texts in Mathematics]] |volume=13 |edition=2 |publisher=Springer-Verlag |place=New York |year=1992 |pages=x+376 |isbn=0-387-97845-3 |mr=1245487 |doi=10.1007/978-1-4612-4418-9}} * Atiyah, M. F., MacDonald, I. G. (1969). Introduction to commutative algebra. Addison-Wesley-Longman. {{ISBN|978-0-201-40751-8}} * {{cite book |last1=Bourbaki |first1=Nicolas |author-link1=Nicolas Bourbaki |title=Commutative Algebra: Chapters 1-7 |date=1989 |publisher=Springer-Verlag |isbn=978-0-387-19371-7 |url=https://books.google.com/books?id=hO69SgAACAAJ |language=en}} * {{cite book|author-link=David Eisenbud|last1=Eisenbud|first1=David|title=Commutative Algebra with a View Toward Algebraic Geometry|series=Graduate Texts in Mathematics|volume=150|publisher=Springer-Verlag|year=1995|isbn=0-387-94268-8|doi=10.1007/978-1-4612-5350-1}} * {{cite journal | last1 = Formanek | first1 = Edward | author1-link=Edward W. Formanek | last2 = Jategaonkar | first2 = Arun Vinayak | date = 1974 | title = Subrings of Noetherian rings | url = https://www.ams.org/journals/proc/1974-046-02/S0002-9939-1974-0414625-5/home.html | journal = [[Proceedings of the American Mathematical Society]] | volume = 46 | issue = 2 | pages = 181–186 | doi = 10.2307/2039890 | jstor = 2039890 | doi-access= free }} * {{Citation|last1=Hotta|first1=Ryoshi|last2=Takeuchi|first2=Kiyoshi|last3=Tanisaki|first3=Toshiyuki|title=D-modules, perverse sheaves, and representation theory|series=Progress in Mathematics|volume=236|publisher=Birkhäuser|year=2008|isbn=978-0-8176-4363-8|mr=2357361|doi=10.1007/978-0-8176-4523-6| zbl=1292.00026}} * {{Cite book|title = A first course in noncommutative rings|last = Lam|first = Tsit Yuen|author-link=Tsit Yuen Lam|publisher = Springer|year = 2001|isbn = 0387951830|location = New York|pages = 19|edition=2nd|series= Graduate Texts in Mathematics|volume= 131|doi=10.1007/978-1-4419-8616-0|mr=1838439 }} * Chapter X of {{Lang Algebra|edition=3}} * {{Citation | last1=Matsumura | first1=Hideyuki | title=Commutative Ring Theory | publisher=[[Cambridge University Press]] | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-36764-6 | year=1989}} ==External links== * {{springer|title=Noetherian ring|id=p/n066850}} [[Category:Ring theory]]
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