Jacobson ring - Revision history

Vipul: /* Definition */

2013-07-18T15:55:29Z

Definition

← Older revision		Revision as of 15:55, 18 July 2013
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	==Definition==		==Definition==

	~~===Symbol-free definition===~~

	The following are equivalent definitions of Jacobson ring.		The following are equivalent definitions of Jacobson ring.

Vipul: /* Metaproperties */

2013-07-18T15:55:10Z

Metaproperties

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	A [[local domain]] that is not a field is not Jacobson. More generally, any [[local ring]] that has prime ideals other than the maximal ideal is not Jacobson.		A [[local domain]] that is not a field is not Jacobson. More generally, any [[local ring]] that has prime ideals other than the maximal ideal is not Jacobson.

	~~==Metaproperties==~~

	~~{{Q-closed curing property}}~~

	A quotient of a Jacobson ring by any ideal is also a Jacobson ring. This is clear, for instance, if we use the characterization in terms of every quotient ring having equal nilradical and Jacobson radical.{{proofat\|[[Jacobson is quotient-closed]]}}

	~~{{poly-closed curing property}}~~

	If <math>R</math> is a Jacobson ring, so is the [[polynomial ring]] <math>R[x]</math>. This is an important observation that forms part of the proof of the [[Hilbert nullstellensatz]] (where the starting ring, <math>R</math> is a field and hence clearly a Jacobson ring).

	~~{{proofat\|[[Jacobson is polynomial-closed]]}}~~

Vipul: /* Relation with other properties */

2013-07-18T15:54:53Z

Relation with other properties

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

	{\| class="sortable"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 at 15:54, 18 July 2013

2013-07-18T15:54:36Z

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	The equivalence with the fourth condition is termed [[Rabinowitch's trick]].		The equivalence with the fourth condition is termed [[Rabinowitch's trick]].

			==Metaproperties==

			{\| class="sortable" border="1"
			! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
			\|-
			\| [[satisfies metaproperty::polynomial-closed property of commutative unital rings]] \|\| Yes \|\| [[Jacobson is polynomial-closed]] \|\| Suppose <math>R</math> is a Jacobson ring. Then, the [[polynomial ring]] <math>R[x]</math> is also a Jacobson ring.
			\|-
			\| [[satisfies metaproperty::quotient-closed property of commutative unital rings]] \|\| Yes \|\| [[Jacobson is quotient-closed]] \|\| If <math>R</math> is a Jacobson ring and <math>I</math> is an [[ideal]] in <math>R</math>, then the [[quotient ring]] <math>R/I</math> is also a Jacobson ring.
			\|}
	==Relation with other properties==		==Relation with other properties==

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

	* [[~~Polynomial~~ ring over a field]]		{\| class="sortable"border="1"
	* [[~~Field~~]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Artinian ring]]		\|-
	* [[~~Zero~~-dimensional ring]]		\| [[Weaker than::polynomial ring over a field]] \|\| [[polynomial ring]] of the form <math>K[x]</math> where <math>K</math> is a [[field]]. \|\| \|\| \|\| {{intermeidate notions short\|Jacobson ring\|polynomial ring over a field}}
			\|-
			\| [[Weaker than::field]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Jacobson ring\|field}}
			\|-
			\| [[Weaker than::Artinian ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Jacobson ring\|Artinian ring}}
			\|-
			\| [[Weaker than::zero-dimensional ring]] \|\| every [[prime ideal]] is a [[maximal ideal]] \|\| \|\| \|\| {{intermediate notions short\|Jacobson ring\|zero-dimensional ring}}
			\|-
			\| [[Weaker than::multivariate polynomial ring over a field]] \|\| of the form <math>K[x_1,x_2,\dots,x_n]</math> where <math>K</math> is a [[field]]. \|\| \|\| \|\| {{intermediate notions short\|Jacobson ring\|multivariate polynomial ring over a field}}
			\|}

			===Opposite properties===

			A [[local domain]] that is not a field is not Jacobson. More generally, any [[local ring]] that has prime ideals other than the maximal ideal is not Jacobson.

	==Metaproperties==		==Metaproperties==

Vipul at 15:47, 18 July 2013

2013-07-18T15:47:45Z

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	\| 5 \|\| localization a field implies a field \|\| for any [[prime ideal]], if the quotient contains an element at which its localization is a field, then the quotient is itself a field. \|\| for any prime ideal <math>P</math> and any <math>x \in R/P</math>, consider the [[localization at a multiplicative subset\|localization]] of <math>R/P</math> at the set of powers of <math>x</math>. If the localization is a field, then <math>R/P</math> must itself be a field.		\| 5 \|\| localization a field implies a field \|\| for any [[prime ideal]], if the quotient contains an element at which its localization is a field, then the quotient is itself a field. \|\| for any prime ideal <math>P</math> and any <math>x \in R/P</math>, consider the [[localization at a multiplicative subset\|localization]] of <math>R/P</math> at the set of powers of <math>x</math>. If the localization is a field, then <math>R/P</math> must itself be a field.
	\|}		\|}

	~~===Definition with symbols===~~

	~~{{fillin}}~~

	===Equivalence of definitions===		===Equivalence of definitions===

Vipul at 15:47, 18 July 2013

2013-07-18T15:47:16Z

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	===Symbol-free definition===		===Symbol-free definition===

	~~A [[commutative unital ring]] is termed a '''~~Jacobson ring~~''' or a '''Hilbert ring''' if it satisfies the following equivalent conditions:~~		The following are equivalent definitions of Jacobson ring.

	* Every [[prime ideal]] in it is an intersection of [[maximal ideal]]s		{\| class="sortable" border="1"
	* Every [[radical ideal]] in it is an intersection of maximal ideals		! No. !! Shorthand !! A commutative unital ring is termed a Jacobson ring or Hilbert ring if ... !! A commutative unital ring <math>R</math> is termed a Jacobson ring or Hilbert ring if ...
	* In the [[spectrum of a commutative unital ring\|spectrum]], the set of closed points in any closed set is dense		\|-
	* For every [[quotient ring]], the [[nilradical]] equals the [[Jacobson radical]]		\| 1 \|\| prime as intersection of maximals \|\| every [[defining ingredient::prime ideal]] in it is an intersection of [[defining ingredient::maximal ideal]]s. \|\| for every prime ideal <math>P</math> of <math>R</math>, <math>P = \bigcap_{i \in I} M_i</math> where <math>M_i, i \in I</math> is the set of all maximal ideals of <math>R</math> containing <math>P</math>.
	* For any [[prime ideal]], if the quotient contains an element at which its localization is a field, then the quotient is itself a field		\|-
			\| 2 \|\| radical as intersection of maximals \|\| every [[radical ideal]] in it is an intersection of maximal ideals \|\| for every radical ideal <math>J</math> of <math>R</math>, <math>J = \bigcap_{i \in I} M_i</math> where <math>M_i, i \in I</math> is the set of all maximal ideals of <math>R</math> containing <math>J</math>.
			\|-
			\| 3 \|\| spectrum: closed points dense \|\| in the [[spectrum of a commutative unital ring\|spectrum]], the set of closed points in any closed set is dense \|\| in the spectrum <math>\operatorname{Spec}(R)</math>, any closed subset <math>C</math> has the property that the set of closed points in <math>C</math> is a dense subset of <math>C</math>.
			\|-
			\| 4 \|\| quotient: nilradical equals Jacobson ring \|\| for every [[quotient ring]], the [[defining ingredient::nilradical]] equals the [[defining ingredient::Jacobson radical]] \|\| for every ideal <math>I</math>, the quotient ring <math>R/I</math> has the property that the nilradical of <math>R/I</math> (i.e., the set of nilpotent elements, or equivalently, the intersection of all prime ideals) equals the Jacobson radical (the set of elements such that 1 + any multiple of the element is invertible, or equivalently, the intersection of all maximal ideals).
			\|-
			\| 5 \|\| localization a field implies a field \|\| for any [[prime ideal]], if the quotient contains an element at which its localization is a field, then the quotient is itself a field. \|\| for any prime ideal <math>P</math> and any <math>x \in R/P</math>, consider the [[localization at a multiplicative subset\|localization]] of <math>R/P</math> at the set of powers of <math>x</math>. If the localization is a field, then <math>R/P</math> must itself be a field.
			\|}

	===Definition with symbols===		===Definition with symbols===
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	The equivalence of the first three definition follows from the definitions of the terms involved. In particular, it uses the fact that in any commutative unital ring, any [[radical ideal]] is an intersection of [[prime ideal]]s.		The equivalence of the first three definition follows from the definitions of the terms involved. In particular, it uses the fact that in any commutative unital ring, any [[radical ideal]] is an intersection of [[prime ideal]]s.

	The equivalence with the fourth condition is termed [[Rabinowitch's trick]]		The equivalence with the fourth condition is termed [[Rabinowitch's trick]].


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

Vipul: 16 revisions

2008-05-12T16:24:20Z

16 revisions

← Older revision	Revision as of 16:24, 12 May 2008
(No difference)

Vipul: /* Metaproperties */

2008-01-11T17:11:42Z

Metaproperties

← Older revision		Revision as of 17:11, 11 January 2008
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	{{Q-closed curing property}}		{{Q-closed curing property}}

	A quotient of a Jacobson ring by any ideal is also a Jacobson ring. {{proofat\|[[Jacobson is quotient-closed]]}}		A quotient of a Jacobson ring by any ideal is also a Jacobson ring. This is clear, for instance, if we use the characterization in terms of every quotient ring having equal nilradical and Jacobson radical.{{proofat\|[[Jacobson is quotient-closed]]}}

	{{poly-closed curing property}}		{{poly-closed curing property}}

Vipul: /* Symbol-free definition */

2008-01-11T17:11:08Z

Symbol-free definition

← Older revision		Revision as of 17:11, 11 January 2008
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	* Every [[radical ideal]] in it is an intersection of maximal ideals		* Every [[radical ideal]] in it is an intersection of maximal ideals
	* In the [[spectrum of a commutative unital ring\|spectrum]], the set of closed points in any closed set is dense		* In the [[spectrum of a commutative unital ring\|spectrum]], the set of closed points in any closed set is dense
			* For every [[quotient ring]], the [[nilradical]] equals the [[Jacobson radical]]
	* For any [[prime ideal]], if the quotient contains an element at which its localization is a field, then the quotient is itself a field		* For any [[prime ideal]], if the quotient contains an element at which its localization is a field, then the quotient is itself a field

Vipul at 01:47, 10 January 2008

2008-01-10T01:47:05Z

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	{{curing property}}		{{curing property}}

			{{spectrum-determined curing property\|set of closed points in any closed subset is dense}}

	==History==		==History==