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| In [[mathematics]], more specifically in the area of [[Abstract algebra|modern algebra]] known as [[Ring (mathematics)|ring theory]], a '''Noetherian ring''', named after [[Emmy Noether]], is a ring in which every non-empty set of [[ideal (ring theory)|ideal]]s has a maximal element. Equivalently, a ring is Noetherian if it satisfies the [[ascending chain condition]] on ideals; that is, given any chain:
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| :<math>I_1\subseteq\cdots \subseteq I_{k-1}\subseteq I_{k}\subseteq I_{k+1}\subseteq\cdots</math>
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| there exists an ''n'' such that:
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| :<math>I_{n}=I_{n+1}=\cdots.</math>
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| There are other equivalent formulations of the definition of a '''Noetherian ring''' and these are outlined later in the article.
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| The notion of a Noetherian ring is of fundamental importance in both [[Commutative ring|commutative]] and [[noncommutative ring]] theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of [[integer]]s and the [[polynomial ring]] over a [[Field (mathematics)|field]] are both Noetherian rings, and consequently, such theorems as the [[Lasker–Noether theorem]], the [[Krull intersection theorem]], and the [[Hilbert's basis theorem]] hold for them. Furthermore, if a ring is Noetherian, then it satisfies the [[descending chain condition]] on ''[[prime ideal]]s''. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the [[Krull dimension]].
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| == Characterizations ==
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| For [[noncommutative ring]]s, it is necessary to distinguish between three very similar concepts:
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| * A ring is '''left-Noetherian''' if it satisfies the ascending chain condition on left ideals.
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| * A ring is '''right-Noetherian''' if it satisfies the ascending chain condition on right ideals.
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| * A ring is '''Noetherian''' if it is both left- and right-Noetherian.
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| 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.
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| There are other, equivalent, definitions for a ring ''R'' to be left-Noetherian:
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| * Every left ideal ''I'' in ''R'' is [[Finitely-generated module|finitely generated]], i.e. there exist elements ''a''<sub>1</sub>, ..., ''a<sub>n</sub>'' in ''I'' such that ''I'' = ''Ra''<sub>1</sub> + ... + ''Ra''<sub>''n''</sub>.<ref name=":0">Lam (2001), p. 19</ref>
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| * Every [[non-empty]] set of left ideals of ''R'', partially ordered by inclusion, has a [[maximal element]] with respect to [[subset|set inclusion]].<ref name=":0" />
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| Similar results hold for right-Noetherian rings.
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| For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. (The result is due to [[I. S. Cohen]].)
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| == Hilbert's basis theorem ==
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| {{Main|Hilbert's basis theorem}}
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| If ''R'' is a ring, let ''R''[''X''] denote the ring of polynomials in the indeterminate ''X'' over ''R''. [[David Hilbert|Hilbert]] proved that if ''R'' is "not too large", in the sense that if ''R'' is Noetherian, the same must be true for ''R''[''X'']. Formally,
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| <blockquote>'''Hilbert's Basis Theorem.''' If ''R'' is a Noetherian ring, then ''R''[''X''] is a Noetherian ring.</blockquote>
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| <blockquote>'''Corollary.''' If ''R'' is a Noetherian ring, then ''R''[''X''<sub>1</sub>, ..., ''X<sub>n</sub>''] is a Noetherian ring.</blockquote>
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| For a proof of this result, see the [[Hilbert%27s_basis_theorem#Proof|corresponding section]] on the Hilbert's basis theorem page. Geometrically, the result asserts that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set (the solution set of a collection of polynomials in ''n'' variables is generally a geometric object (such as a curve or a surface) in ''n''-space).
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| == Primary decomposition ==
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| {{Main|Lasker–Noether theorem}}
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| In the ring '''Z''' of integers, an arbitrary ideal is of the form (''n'') for some integer ''n'' (where (''n'') denotes the set of all integer multiples of ''n''). If ''n'' is non-zero, and is neither 1 nor −1, by the [[fundamental theorem of arithmetic]], there exist primes ''p<sub>i</sub>'', and positive integers ''e<sub>i</sub>'', with <math>n=\prod_{i} {p_i}^{e_i}</math>. In this case, the ideal (''n'') may be written as the intersection of the ideals (''p<sub>i</sub><sup>e<sub>i</sub></sup>''); that is, <math>(n)=\cap_{i} ({p_i}^{e_i})</math>. This is referred to as a ''primary decomposition'' of the ideal (''n'').
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| In general, an ideal ''Q'' of a ring is said to be ''[[Primary ideal|primary]]'' if ''Q'' is [[proper subset|proper]] and whenever ''xy'' ∈ ''Q'', either ''x'' ∈ ''Q'' or ''y<sup>n</sup>'' ∈ ''Q'' for some positive integer ''n''. In '''Z''', the primary ideals are precisely the ideals of the form (''p<sup>e</sup>'') where ''p'' is prime and ''e'' is a positive integer. Thus, a primary decomposition of (''n'') corresponds to representing (''n'') as the intersection of finitely many primary ideals.
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| Since the fundamental theorem of arithmetic applied to a non-zero integer ''n'' that is neither 1 nor −1 also asserts uniqueness of the representation <math>n=\prod_{i} {p_i}^{e_i}</math> for ''p<sub>i</sub>'' prime and ''e<sub>i</sub>'' positive, a primary decomposition of (''n'') is essentially ''unique''.
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| For all of the above reasons, the following theorem, referred to as the ''[[Lasker–Noether theorem]]'', may be seen as a certain generalization of the fundamental theorem of arithmetic:
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| <blockquote>'''Lasker-Noether Theorem.''' Let ''R'' be a commutative Noetherian ring and let ''I'' be an ideal of ''R''. Then ''I'' may be written as the intersection of finitely many primary ideals with distinct [[Radical of an ideal|radicals]]; that is:
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| : <math>I=\bigcap_{i=1}^t Q_i</math>
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| with ''Q<sub>i</sub>'' primary for all ''i'' and Rad(''Q<sub>i</sub>'') ≠ Rad(''Q<sub>j</sub>'') for ''i'' ≠ ''j''. Furthermore, if:
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| : <math>I=\bigcap_{i=1}^k P_i</math>
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| is decomposition of ''I'' with Rad(''P<sub>i</sub>'') ≠ Rad(''P<sub>j</sub>'') for ''i'' ≠ ''j'', and both decompositions of ''I'' are ''irredundant'' (meaning that no proper subset of either {''Q''<sub>1</sub>, ..., ''Q<sub>t</sub>''} or {''P''<sub>1</sub>, ..., ''P<sub>k</sub>''} yields an intersection equal to ''I''), ''t'' = ''k'' and (after possibly renumbering the ''Q<sub>i</sub>'') Rad(''Q<sub>i</sub>'') = Rad(''P<sub>i</sub>'') for all ''i''.</blockquote>
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| For any primary decomposition of ''I'', the set of all radicals, that is, the set {Rad(''Q''<sub>1</sub>), ..., Rad(''Q<sub>t</sub>'')} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the [[associated prime|assassinator]] of the module ''R''/''I''; that is, the set of all [[annihilator (ring theory)|annihilators]] of ''R''/''I'' (viewed as a module over ''R'') that are prime.
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| == Uses ==
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| The Noetherian property is central in [[ring theory]] and in areas that make heavy use of rings, such as [[algebraic geometry]]. The reason behind this is that the Noetherian property is in some sense the ring-theoretic analogue of finiteness. For example, the fact that polynomial rings over a field are Noetherian allows one to prove that any infinite set of polynomial equations can be replaced with a finite set with the same solutions.
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| [[Krull's principal ideal theorem]] states that every [[principal ideal]] in a commutative Noetherian ring has [[height (ring theory)|height]] one; that is, every principal ideal is contained in a [[prime ideal]] minimal amongst nonzero prime ideals. This early result was the first to suggest that Noetherian rings possessed a deep theory of [[Krull dimension|dimension]].
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| See also: [[Artin–Rees lemma]].
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| == Examples ==
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| * Any field, including fields of [[rational number]]s, [[real number]]s, and [[complex number]]s. (A field only has two ideals — itself and (0).)
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| * Any [[principal ideal domain]], such as the [[integers]], is Noetherian since every ideal is generated by a single element.
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| * A [[Dedekind domain]] (e.g., [[ring of integers|rings of integers]]) is Noetherian since every ideal is generated by at most two elements. The "Noetherian" follows from the [[Krull–Akizuki theorem]]. The bounds on the number of the generators is a corollary of the [[Forster–Swan theorem]] (or basic ring theory).
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| * The [[coordinate ring]] of an affine variety is a noetherian ring, as a consequence of the Hilbert basis theorem.
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| * 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; thus, noetherian.<ref>{{harvnb|Bourbaki|1989|loc=Ch III, §2, no. 10, Remarks at the end of the number}}</ref>
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| * The ring of polynomials in finitely-many variables over the integers or a field.
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| Rings that are not Noetherian tend to be (in some sense) very large. Here are three examples of non-Noetherian rings:
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| * 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.
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| * The ring of [[algebraic integers]] is not Noetherian. For example, it contains the infinite ascending chain of principal ideals: (2), (2<sup>1/2</sup>), (2<sup>1/3</sup>), (2<sup>1/4</sup>), ...
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| * The ring of continuous functions 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.
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| However, a non-Noetherian ring can be a subring of a Noetherian ring: trivially because any integral domain is a subring of a field. For a less trivial,
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| * The ring of rational functions 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.
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| Indeed, there are rings that are left Noetherian, but not right Noetherian, so that one must be careful in measuring the "size" of a ring this way. <!-- give matrix ring example, [Z,Q;0,Q] I believe-->
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| A [[unique factorization domain]] is not necessarily a noetherian ring. It does satisfy a weaker condition: the [[ascending chain condition on principal ideals]].
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| == Properties ==
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| * If ''R'' is a Noetherian ring, then ''R''[''X''] is Noetherian by the [[Hilbert basis theorem]]. Also, ''R''<nowiki>[[</nowiki>''X''<nowiki>]]</nowiki>, the [[Formal power series|power series ring]] is a Noetherian ring.
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| * If ''R'' is a Noetherian ring and ''I'' is a two-sided ideal, then the [[factor ring]] ''R''/''I'' is also Noetherian.
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| * Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. (This follows from the two previous properties.)
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| * Every [[Localization of a ring|localization]] of a commutative Noetherian ring is Noetherian.
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| * 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 right Artinian. The analogous statements with "right" and "left" interchanged are also true.
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| * A ring ''R'' is left-Noetherian if and only if every finitely generated [[module (mathematics)|left ''R''-module]] is a [[Noetherian module]].
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| * A left Noetherian ring is left [[coherent ring|coherent]] and a left Noetherian [[Domain (ring theory)|domain]] is a left [[Ore domain]].
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| * A ring is (left/right) Noetherian if and only if every direct sum of [[injective module|injective (left/right) modules]] is injective. Every injective module can be decomposed as direct sum of indecomposable injective modules.
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| * In a commuative Noetherian ring, there are only finitely many [[minimal prime ideal]]s.
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| == See also ==
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| *[[Krull–Akizuki theorem]]
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| *[[Noetherian scheme]]
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| *[[Artinian ring]]
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| ==References==
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| {{reflist}}
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| * [[Nicolas Bourbaki]], Commutative algebra
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| * Chapter X of {{Lang Algebra|edition=3}}
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| * {{Cite book|title = A first course in noncommutative rings|last = Lam|first = T.Y.|publisher = Springer|year = 2001|isbn = 0387951830|location = New York|pages = 19}}
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| ==External links==
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| * {{springer|title=Noetherian ring|id=p/n066850}}
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| [[Category:Ring theory]]
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