Noetherian ring - Revision history

Vipul at 03:01, 18 July 2013

2013-07-18T03:01:21Z

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 {{further|[[equivalence of definitions of Noetherian ring]]}}
 ==Relation with other properties==
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 |-
 | [[Stronger than::Coherent ring]] || || || || {{intermediate notions short|Coherent ring|Noetherian ring}}
 |}

Vipul: /* Metaproperties */

2013-07-18T03:00:58Z

Metaproperties

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	==Metaproperties==		==Metaproperties==

	{~~{poly~~-closed ~~curing~~ property}}		{\| class="sortable" border="1"
			! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
	~~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]]}}		\| [[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.
			\|-
	~~{{Q-closed curing property}}~~		\| [[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.
			\|-
	~~The~~ [[quotient ~~ring~~]] of a Noetherian ring by an ideal, is also Noetherian. ~~{{further~~\|[[~~Noetherianness is quotient~~-closed]]}}		\| [[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.
			\|-
	{{not S-closed ~~curing property}}~~		\| [[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.
			\|-
	~~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.		\| [[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.
			\|-
	{{localization-closed ~~curing~~ property}}		\| [[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.
			\|}
	~~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]]}~~}

Vipul at 02:33, 18 July 2013

2013-07-18T02:33:38Z

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	! No. !! Shorthand !! A [[commutative unital ring]] is termed Noetherian if ... !! A [[commutative unital ring]] <math>R</math> is termed Noetherian if ...		! No. !! Shorthand !! A [[commutative unital ring]] is termed Noetherian if ... !! A [[commutative unital ring]] <math>R</math> is termed Noetherian if ...
	\|-		\|-
	\| [[ascending chain condition]] on ideals \|\| any ascending chain of ideals stabilizes after a finite length \|\| '''arbitrary version''': if <math>J</math> is a well-ordered set and <math>I_j, j \in J</math> are ideals such that <math>I_j \subseteq I_k</math> for <math>j \le k</math>, there exists some <math>j \in J</math> such that <math>I_j = I_k</math> for all <math>k \ge j</math>.<br>'''countable chain version''': If <math>I_1 \subseteq I_2 \subseteq \dots </math> is an ascending chain of ideals, then there exists <math>I_n</math> such that <math>I_n = I_m</math> for all <math>m \ge n</math>.		\| 1 \|\| [[ascending chain condition]] on ideals \|\| any ascending chain of ideals stabilizes after a finite length \|\| '''arbitrary version''': if <math>J</math> is a well-ordered set and <math>I_j, j \in J</math> are ideals such that <math>I_j \subseteq I_k</math> for <math>j \le k</math>, there exists some <math>j \in J</math> such that <math>I_j = I_k</math> for all <math>k \ge j</math>.<br>'''countable chain version''': If <math>I_1 \subseteq I_2 \subseteq \dots </math> is an ascending chain of ideals, then there exists <math>I_n</math> such that <math>I_n = I_m</math> for all <math>m \ge n</math>.
	\|-		\|-
	\| finite generation of ideals \|\| every [[ideal]] in the ring is [[defining ingredient::finitely generated ideal\|finitely generated]]. \|\| for every ideal <math>I</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in I</math> such that <math>I</math> is the ideal generated by <math>x_1, x_2, \dots, x_n</math>.		\| 2 \|\| finite generation of ideals \|\| every [[ideal]] in the ring is [[defining ingredient::finitely generated ideal\|finitely generated]]. \|\| for every ideal <math>I</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in I</math> such that <math>I</math> is the ideal generated by <math>x_1, x_2, \dots, x_n</math>.
	\|-		\|-
	\| finite generation of prime ideals \|\| every [[defining ingredient::prime ideal]] in the ring is [[finitely generated ideal\|finitely generated]]. \|\| for every prime ideal <math>P</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in P</math> such that <math>P</math> equals the ideal generated by <math>x_1, x_2, \dots, x_n</math>.		\| 3 \|\| finite generation of prime ideals \|\| every [[defining ingredient::prime ideal]] in the ring is [[finitely generated ideal\|finitely generated]]. \|\| for every prime ideal <math>P</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in P</math> such that <math>P</math> equals the ideal generated by <math>x_1, x_2, \dots, x_n</math>.
	\|}		\|}

Vipul at 02:33, 18 July 2013

2013-07-18T02:33:20Z

← Older revision		Revision as of 02:33, 18 July 2013
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	==Definition==		==Definition==

	~~===Symbol-free definition===~~		Below are some '''equivalent''' definitions of Noetherian ring:

	A [[commutative unital ring]] is termed Noetherian if ~~it satisfies~~ the ~~following equivalent conditions~~:		{\| class="sortable" border="1"
			! No. !! Shorthand !! A [[commutative unital ring]] is termed Noetherian if ... !! A [[commutative unital ring]] <math>R</math> is termed Noetherian if ...
			\|-
			\| [[ascending chain condition]] on ideals \|\| any ascending chain of ideals stabilizes after a finite length \|\| '''arbitrary version''': if <math>J</math> is a well-ordered set and <math>I_j, j \in J</math> are ideals such that <math>I_j \subseteq I_k</math> for <math>j \le k</math>, there exists some <math>j \in J</math> such that <math>I_j = I_k</math> for all <math>k \ge j</math>.<br>'''countable chain version''': If <math>I_1 \subseteq I_2 \subseteq \dots </math> is an ascending chain of ideals, then there exists <math>I_n</math> such that <math>I_n = I_m</math> for all <math>m \ge n</math>.
			\|-
			\| finite generation of ideals \|\| every [[ideal]] in the ring is [[defining ingredient::finitely generated ideal\|finitely generated]]. \|\| for every ideal <math>I</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in I</math> such that <math>I</math> is the ideal generated by <math>x_1, x_2, \dots, x_n</math>.
			\|-
			\| finite generation of prime ideals \|\| every [[defining ingredient::prime ideal]] in the ring is [[finitely generated ideal\|finitely generated]]. \|\| for every prime ideal <math>P</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in P</math> such that <math>P</math> equals the ideal generated by <math>x_1, x_2, \dots, x_n</math>.
			\|}

	* [[Ascending chain condition]] on ideals: Any ascending chain of ~~ideals stabilizes after a finite length.~~		===Equivalence of definitions===
	* Every [[ideal]] in the ring is [[defining ingredient::finitely generated ideal\|finitely generated]].
	* Every [[prime ideal]] in the ring is finitely generated.

	~~===Definition with symbols===~~		{{further\|[[equivalence of definitions of Noetherian ring]]}}

	A [[~~commutative unital~~ ring]] ~~<math>R</math> is termed a '''Noetherian ring''' if it satisfies the following equivalent conditions:~~

	* If <math>J</math> is a well-ordered set and <math>I_j, j \in J</math> are ideals such that <math>I_j \subseteq I_k</math> for <math>j \le k</math>, there exists some <math>j \in J</math> such that <math>I_j = I_k</math> for all <math>k \ge j</math>.
	* If <math>I_1 \subseteq I_2 \subseteq \dots </math> is an ascending chain of ideals, then there exists <math>I_n</math> such that <math>I_n = I_m</math> for all <math>m \ge n</math>.
	* For every ideal <math>I</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in I</math> such that <math>I</math> is the ideal generated by <math>x_1, x_2, \dots, x_n</math>.
	* For every prime ideal <math>P</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in P</math> such that <math>P</math> equals the ideal generated by <math>x_1, x_2, \dots, x_n</math>.

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Weaker properties */

2013-07-18T02:28:27Z

Weaker properties

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	===Weaker properties===		===Weaker properties===

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-

Vipul: /* Weaker properties */

2013-07-18T02:28:18Z

Weaker properties

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	{\| class="wikitable" border="1"		{\| class="wikitable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::Coherent ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Coherent ring\|Noetherian ring}}		\| [[Stronger than::Coherent ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Coherent ring\|Noetherian ring}}

Vipul: /* Conjunction with other properties */

2013-07-18T02:27:45Z

Conjunction with other properties

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	===Conjunction with other properties===		===Conjunction with other properties===

	* [[Weaker than::Noetherian domain]] ~~is a Noetherian ring that is also an~~ [[integral domain]].		{\| class="sortable" border="1"
	* [[Weaker than::~~Reduced~~ Noetherian ring]] ~~is a Noetherian ring that is also a~~ [[reduced ring]]: it has no nonzero nilpotent elements.		! Conjunction !! Other component of conjunction !! Additional comments
	* [[Weaker than::Noetherian normal domain]] ~~is a Noetherian ring that is also a~~ [[normal domain]].		\|-
	* [[Weaker than::Noetherian unique factorization domain]]		\| [[Weaker than::Noetherian domain]] \|\| [[integral domain]] \|\|
	* [[Weaker than::~~Local~~ Noetherian ring]] ~~is a Noetherian ring that is also a~~ [[local ring]].		\|-
	* [[Weaker than::~~Local~~ Noetherian domain]] ~~is a Noetherian ring that is also a~~ [[local domain]].		\| [[Weaker than::reduced Noetherian ring]] \|\| [[reduced ring]]: it has no nonzero nilpotent elements. \|\|
	* [[Weaker than::~~Zero~~-dimensional Noetherian ring]] ~~is a Noetherian ring that is also a~~ [[zero-dimensional ring]]: every [[prime ideal]] in it is a [[maximal ideal]].		\|-
	* [[Weaker than::~~One~~-dimensional Noetherian domain]] ~~is a Noetherian ring that is also a~~ [[one-dimensional domain]].		\| [[Weaker than::Noetherian normal domain]] \|\| [[normal domain]] \|\|
	* [[Weaker than::~~Finite~~-dimensional Noetherian ring]] ~~is a Noetherian ring that is also a~~ [[finite-dimensional ring]]: its [[Krull dimension]] is finite.		\|-
			\| [[Weaker than::Noetherian unique factorization domain]] \|\| [[unique factorization domain]] \|\|
			\|-
			\| [[Weaker than::local Noetherian ring]]\|\| [[local ring]] \|\|
			\|-
			\| [[Weaker than::local Noetherian domain]] \|\| [[local domain]] \|\|
			\|-
			\| [[Weaker than::zero-dimensional Noetherian ring]] \|\| [[zero-dimensional ring]]: every [[prime ideal]] in it is a [[maximal ideal]] \|\|
			\|-
			\| [[Weaker than::one-dimensional Noetherian domain]] \|\| [[one-dimensional domain]]: [[integral domain]] in which every nonzero prime ideal is maximal \|\|
			\|-
			\| [[Weaker than::finite-dimensional Noetherian ring]] \|\| [[finite-dimensional ring]]: its [[Krull dimension]] is finite. \|\|
			\|}

	===Stronger properties===		===Stronger properties===

Vipul: /* Definition */

2011-12-19T00:08:04Z

Definition

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	* If <math>J</math> is a well-ordered set and <math>I_j, j \in J</math> are ideals such that <math>I_j \subseteq I_k</math> for <math>j \le k</math>, there exists some <math>j \in J</math> such that <math>I_j = I_k</math> for all <math>k \ge j</math>.		* If <math>J</math> is a well-ordered set and <math>I_j, j \in J</math> are ideals such that <math>I_j \subseteq I_k</math> for <math>j \le k</math>, there exists some <math>j \in J</math> such that <math>I_j = I_k</math> for all <math>k \ge j</math>.
	* If <math>I_1 \subseteq I_2 \subseteq \dots </math> is an ascending chain of ideals, then there exists <math>I_n</math> such that <math>~~I_N~~ = I_m</math> for all <math>m \ge n</math>.		* If <math>I_1 \subseteq I_2 \subseteq \dots </math> is an ascending chain of ideals, then there exists <math>I_n</math> such that <math>I_n = I_m</math> for all <math>m \ge n</math>.
	* For every ideal <math>I</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in I</math> such that <math>I</math> is the ideal generated by <math>x_1, x_2, \dots, x_n</math>.		* For every ideal <math>I</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in I</math> such that <math>I</math> is the ideal generated by <math>x_1, x_2, \dots, x_n</math>.
	* For every prime ideal <math>P</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in P</math> such that <math>P</math> equals the ideal generated by <math>x_1, x_2, \dots, x_n</math>.		* For every prime ideal <math>P</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in P</math> such that <math>P</math> equals the ideal generated by <math>x_1, x_2, \dots, x_n</math>.

Vipul: /* Stronger properties */

2011-12-19T00:01:59Z

Stronger properties

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	===Stronger properties===		===Stronger properties===

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::Polynomial ring over a field]] \|\| <math>k[x]</math> for a field <math>k</math> \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|polynomial ring over a field}}		\| [[Weaker than::Polynomial ring over a field]] \|\| <math>k[x]</math> for a field <math>k</math> \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|polynomial ring over a field}}

Vipul: /* Definition with symbols */

2011-12-19T00:00:52Z

Definition with symbols

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	A [[commutative unital ring]] <math>R</math> is termed a '''Noetherian ring''' if it satisfies the following equivalent conditions:		A [[commutative unital ring]] <math>R</math> is termed a '''Noetherian ring''' if it satisfies the following equivalent conditions:

	* If <math>J</math> is a well-ordered set and <math>I_j, j \in J</math> are ideals such that <math>~~I_J~~ \subseteq I_k</math> for <math>j \le k</math>, there exists some <math>j \in J</math> such that <math>I_j = I_k</math> for all <math>k \ge j</math>.		* If <math>J</math> is a well-ordered set and <math>I_j, j \in J</math> are ideals such that <math>I_j \subseteq I_k</math> for <math>j \le k</math>, there exists some <math>j \in J</math> such that <math>I_j = I_k</math> for all <math>k \ge j</math>.
	* If <math>I_1 \subseteq I_2 \subseteq \dots </math> is an ascending chain of ideals, then there exists <math>I_n</math> such that <math>I_N = I_m</math> for all <math>m \ge n</math>.		* If <math>I_1 \subseteq I_2 \subseteq \dots </math> is an ascending chain of ideals, then there exists <math>I_n</math> such that <math>I_N = I_m</math> for all <math>m \ge n</math>.
	* For every ideal <math>I</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in I</math> such that <math>I</math> is the ideal generated by <math>x_1, x_2, \dots, x_n</math>.		* For every ideal <math>I</math> in <math>R</math>, there exist elements <math>x_1, x_2, \dots, x_n \in I</math> such that <math>I</math> is the ideal generated by <math>x_1, x_2, \dots, x_n</math>.