Noetherian ring: Difference between revisions

Latest revision as of 03:01, 18 July 2013

This article is about a standard (though not very rudimentary) definition in commutative algebra. The article text may, however, contain more than just the basic definition
View a complete list of semi-basic definitions on this wiki

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

Below are some equivalent definitions of Noetherian ring:

No.	Shorthand	A commutative unital ring is termed Noetherian if ...	A commutative unital ring $R$ is termed Noetherian if ...
1	ascending chain condition on ideals	any ascending chain of ideals stabilizes after a finite length	arbitrary version: if $J$ is a well-ordered set and $I_{j}, j \in J$ are ideals such that $I_{j} \subseteq I_{k}$ for $j \leq k$ , there exists some $j \in J$ such that $I_{j} = I_{k}$ for all $k \geq j$ . countable chain version: If $I_{1} \subseteq I_{2} \subseteq \dots$ is an ascending chain of ideals, then there exists $I_{n}$ such that $I_{n} = I_{m}$ for all $m \geq n$ .
2	finite generation of ideals	every ideal in the ring is finitely generated.	for every ideal $I$ in $R$ , there exist elements $x_{1}, x_{2}, \dots, x_{n} \in I$ such that $I$ is the ideal generated by $x_{1}, x_{2}, \dots, x_{n}$ .
3	finite generation of prime ideals	every prime ideal in the ring is finitely generated.	for every prime ideal $P$ in $R$ , there exist elements $x_{1}, x_{2}, \dots, x_{n} \in P$ such that $P$ equals the ideal generated by $x_{1}, x_{2}, \dots, x_{n}$ .

Equivalence of definitions

Further information: equivalence of definitions of Noetherian ring

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
polynomial-closed property of commutative unital rings	Yes	Noetherianness is polynomial-closed	Suppose $R$ is a Noetherian ring. Then, the polynomial ring $R [x]$ is also a Noetherian ring.
quotient-closed property of commutative unital rings	Yes	Noetherianness is quotient-closed	Suppose $R$ is a Noetherian ring and $I$ is an ideal in $R$ . Then, the quotient ring $R / I$ is also a Noetherian ring.
subring-closed property of commutative unital rings	No	Noetherianness is not subring-closed	It is possible to have the following: $R$ is a Noetherian ring and $S$ is a (unital) subring of $R$ , but $S$ is not a Noetherian ring. For instance, consider any non-Noetherian integral domain; this is a subring of a field, which is Noetherian.
localization-closed property of commutative unital rings	Yes	Noetherianness is localization-closed	Suppose $R$ is a Noetherian ring and $S$ is a multiplicatively closed subset of $R$ . Then, the localization of $R$ at $S$ is also Noetherian.
finite direct product-closed property of commutative unital rings	Yes	Noetherianness is finite direct product-closed	Suppose $R_{1}, R_{2}, \dots, R_{n}$ are Noetherian rings. Then, the direct product $R_{1} \times R_{2} \times \dots \times R_{n}$ is also a Noetherian ring.
completion-closed property of commutative unital rings	Yes	Noetherianness is completion-closed	Suppose $R$ is a commutative unital ring and $M$ is a maximal ideal in $R$ . The completion of $R$ at $M$ is also Noetherian.

Relation with other properties

Conjunction with other properties

Conjunction	Other component of conjunction	Additional comments
Noetherian domain	integral domain
reduced Noetherian ring	reduced ring: it has no nonzero nilpotent elements.
Noetherian normal domain	normal domain
Noetherian unique factorization domain	unique factorization domain
local Noetherian ring	local ring
local Noetherian domain	local domain
zero-dimensional Noetherian ring	zero-dimensional ring: every prime ideal in it is a maximal ideal
one-dimensional Noetherian domain	one-dimensional domain: integral domain in which every nonzero prime ideal is maximal
finite-dimensional Noetherian ring	finite-dimensional ring: its Krull dimension is finite.

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Polynomial ring over a field	$k [x]$ for a field $k$			click here
Artinian ring	descending chain of ideals stabilizes eventually	Artinian implies Noetherian	Noetherian not implies Artinian	click here
Principal ideal ring	every ideal is principal	principal ideal ring implies Noetherian	Noetherian not implies principal ideal ring	click here
Dedekind domain				click here
Cohen-Macaulay ring				click here
Affine ring				click here

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Coherent ring				click here

@@ Line 19: / Line 19: @@
 {{further|[[equivalence of definitions of Noetherian ring]]}}
+==Metaproperties==
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::polynomial-closed property of commutative unital rings]] || Yes || [[Noetherianness is polynomial-closed]] || Suppose <math>R</math> is a Noetherian ring. Then, the [[polynomial ring]] <math>R[x]</matH> is also a Noetherian ring.
+|-
+| [[satisfies metaproperty::quotient-closed property of commutative unital rings]] || Yes || [[Noetherianness is quotient-closed]] || Suppose <math>R</math> is a Noetherian ring and <math>I</math> is an [[ideal]] in <math>R</math>. Then, the [[quotient ring]] <math>R/I</math> is also a Noetherian ring.
+|-
+| [[dissatisfies metaproperty::subring-closed property of commutative unital rings]] || No || [[Noetherianness is not subring-closed]] || It is possible to have the following: <math>R</math> is a Noetherian ring and <math>S</math> is a (unital) subring of <math>R</math>, but <math>S</math> is not a Noetherian ring. For instance, consider any non-Noetherian [[integral domain]]; this is a subring of a [[field]], which is Noetherian.
+|-
+| [[satisfies metaproperty::localization-closed property of commutative unital rings]] || Yes || [[Noetherianness is localization-closed]] || Suppose <math>R</math> is a Noetherian ring and <math>S</math> is a multiplicatively closed subset of <math>R</math>. Then, the localization of <math>R</math> at <math>S</math> is also Noetherian.
+|-
+| [[satisfies metaproperty::finite direct product-closed property of commutative unital rings]] || Yes || [[Noetherianness is finite direct product-closed]] || Suppose <math>R_1, R_2,\dots,R_n</math> are Noetherian rings. Then, the direct product <math>R_1 \times R_2 \times \dots \times R_n</math> is also a Noetherian ring.
+|-
+| [[satisfies metaproperty::completion-closed property of commutative unital rings]] || Yes || [[Noetherianness is completion-closed]] || Suppose <math>R</matH> is a commutative unital ring and <math>M</math> is a [[maximal ideal]] in <math>R</math>. The completion of <math>R</math> at <math>M</math> is also Noetherian.
+|}
 ==Relation with other properties==
@@ Line 71: / Line 89: @@
 | [[Stronger than::Coherent ring]] || || || || {{intermediate notions short|Coherent ring|Noetherian ring}}
 |}
-==Metaproperties==
-{{poly-closed curing property}}
-The polynomial ring over a Noetherian ring is again Noetherian. This is a general formulation of the [[Hilbert basis theorem]], which asserts in particular that the [[polynomial ring over a field]] is Noetherian.
-{{further|[[Noetherianness is polynomial-closed]]}}
-{{Q-closed curing property}}
-The [[quotient ring]] of a Noetherian ring by an ideal, is also Noetherian. {{further|[[Noetherianness is quotient-closed]]}}
-{{not S-closed curing property}}
-A subring of a Noetherian ring is not necessarily Noetherian. For this, consider any non-Noetherian [[integral domain]]; this is a subring of a [[field]], which is Noetherian.
-{{localization-closed curing property}}
-A localization of a Noetherian ring is Noetherian. Intuitively, when we take localizations, we land up with fewer ideals, so the ascending chain condition becomes easier to satisfy. {{further|[[Noetherianness is localization-closed]]}}
-{{finite DP-closed curing property}}
-A direct product of finitely many Noetherian rings is again Noetherian. This is essentially because ideals in the direct product look like direct products of ideals in the factors. {{proofat|[[Noetherianness is finite direct product-closed]]}}
-{{completion-closed curing property}}
-The completion of a Noetherian ring at a [[maximal ideal]] is again a Noetherian ring. {{proofat|[[Noetherianness is completion-closed]]}}