Prime ideal - Revision history

Vipul: /* Equivalent definitions in tabular format */

2011-12-18T16:57:11Z

Equivalent definitions in tabular format

← Older revision		Revision as of 16:57, 18 December 2011
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	\| 3 \|\| product of finitely many elements version \|\| It is a [[proper ideal]] and whenever the product of finitely many elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal \|\| <math>I</math> is proper in <math>R</math> and whenever <math>a_1,a_2,\ldots,a_n \in R</math> are such that <math>a_1a_2 \ldots a_n \in I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>a_i \in I</math>. Note that the <math>a_i</math> may be equal or distinct, i.e., some may be equal, some may be distinct.		\| 3 \|\| product of finitely many elements version \|\| It is a [[proper ideal]] and whenever the product of finitely many elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal \|\| <math>I</math> is proper in <math>R</math> and whenever <math>a_1,a_2,\ldots,a_n \in R</math> are such that <math>a_1a_2 \ldots a_n \in I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>a_i \in I</math>. Note that the <math>a_i</math> may be equal or distinct, i.e., some may be equal, some may be distinct.
	\|-		\|-
	\| 4 \|\| product of finitely many ideals version \|\| It is a [[proper ideal]] and whenever the product of finitely many ideals is contained in it, one of the ideals is contained in it. \|\| <math>I</math> is proper in <math>R</math> and whenever <math>J_1,J_2,\ldots,J_n</math> are [[ideal]]s in <math>R</math> such that <math>J_1J_2\ldots J_n \~~subset~~ I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>J_i \~~subset~~ I</math>		\| 4 \|\| product of finitely many ideals version \|\| It is a [[proper ideal]] and whenever the product of finitely many ideals is contained in it, one of the ideals is contained in it. \|\| <math>I</math> is proper in <math>R</math> and whenever <math>J_1,J_2,\ldots,J_n</math> are [[ideal]]s in <math>R</math> such that <math>J_1J_2\ldots J_n \subseteq I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>J_i \subseteq I</math>
	\|-		\|-
	\| 5 \|\| complement saturated subset version \|\| It is a [[proper ideal]] and its complement is a multiplicatively closed [[defining ingredient::saturated subset]] (that is, is closed with respect to the operations of multiplication and factorization). \|\| <math>I</math> is proper in <math>R</math> and the complement of <math>I</math> in <math>R</math> is a multiplicatively closed and saturated subset i.e. <math>ab \in R \setminus I \iff a \in R \setminus I</math> or <math>b \in R \setminus I</math>.		\| 5 \|\| complement saturated subset version \|\| It is a [[proper ideal]] and its complement is a multiplicatively closed [[defining ingredient::saturated subset]] (that is, is closed with respect to the operations of multiplication and factorization). \|\| <math>I</math> is proper in <math>R</math> and the complement of <math>I</math> in <math>R</math> is a multiplicatively closed and saturated subset i.e. <math>ab \in R \setminus I \iff a \in R \setminus I</math> or <math>b \in R \setminus I</math>.

Vipul at 16:55, 18 December 2011

2011-12-18T16:55:13Z

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	~~{{basicdef}}~~
	~~{{curing-ideal property}}~~
	~~{{quotient is a\|integral domain}}~~

	==Definition==		==Definition==

	===~~Symbol-free definition===~~		===Equivalent definitions in tabular format===

	~~An [[ideal]]~~ in ~~a [[commutative unital ring]] is termed a '''prime ideal''' if it is [[proper ideal\|proper]], and satisfies the following equivalent conditions:~~

	* Whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal
	* Whenever the [[product of ideals\|product]] of two ideals is contained in it, one of the ideals is contained in it
	* Whenever the product of finitely many elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal
	* Whenever the product of finitely many ideals is contained in it, one of the ideals is contained in it
	* It is an ideal whose complement is a [[saturated subset]] (that is, is closed with respect to the operations of multiplication and factorization).
	* The [[quotient ring]] by that ideal is an [[integral domain]]

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

	An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is termed a ~~'''~~prime ideal~~'''~~ if <math>I</math> is proper in <math>R</math> and ~~it satisfies the following equivalent conditions:~~		{\| class="sortable" border="1"
			! No. !! Shorthand !! An [[ideal]] in a [[commutative unital ring]] is termed a [[prime ideal]] if ... !! An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is termed a [[prime ideal]] if ...
	* Whenever <math>a, b \in R</math> are such that <math>ab \in I</math> then either <math>a \in I</math> or <math>b \in I</math>		\|-
	* Whenever <math>J</math> and <math>K</math> are ideals such that <math>JK \~~subset~~ I</math>, then either <math>J \~~subset~~ I</math> or <math>K \~~subset~~ I</math>		\| 1 \|\| product of two elements version \|\| It is a [[proper ideal]] and whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal \|\| <math>I</math> is proper in <math>R</math> and whenever <math>a, b \in R</math> are (possibly equal, possibly distinct elements) such that <math>ab \in I</math> then either <math>a \in I</math> or <math>b \in I</math> (note: it's also possible that both <math>a,b</math> are in <math>I</math>).
	* Whenever <math>a_1,a_2,\ldots,a_n \in R</math> such that <math>a_1a_2 \ldots a_n \in I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>a_i \in I</math>		\|-
	* Whenever <math>J_1,J_2,\ldots,J_n</math> are [[ideal]]s in <math>R</math> such that <math>J_1J_2\ldots J_n \subset I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>J_i \subset I</math>		\| 2 \|\| product of two ideals version \|\| It is a [[proper ideal]] and whenever the [[product of ideals\|product]] of two ideals is contained in it, one of the ideals is contained in it. \|\| <math>I</math> is proper in <math>R</math> and whenever <math>J</math> and <math>K</math> are (possibly equal, possibly distinct) ideals such that <math>JK \subseteq I</math>, then either <math>J \subseteq I</math> or <math>K \subseteq I</math> (note: it's also possible that both <math>J,K</math> are in <math>I</math>).
	* The complement of <math>I</math> in <math>R</math> is a multiplicatively closed and saturated subset i.e. <math>ab \in R \setminus I \iff a \in R \setminus I</math> or <math>b \in R \setminus I</math>		\|-
	* The quotient ring <math>R/I</math> is an [[integral domain]]		\| 3 \|\| product of finitely many elements version \|\| It is a [[proper ideal]] and whenever the product of finitely many elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal \|\| <math>I</math> is proper in <math>R</math> and whenever <math>a_1,a_2,\ldots,a_n \in R</math> are such that <math>a_1a_2 \ldots a_n \in I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>a_i \in I</math>. Note that the <math>a_i</math> may be equal or distinct, i.e., some may be equal, some may be distinct.
			\|-
			\| 4 \|\| product of finitely many ideals version \|\| It is a [[proper ideal]] and whenever the product of finitely many ideals is contained in it, one of the ideals is contained in it. \|\| <math>I</math> is proper in <math>R</math> and whenever <math>J_1,J_2,\ldots,J_n</math> are [[ideal]]s in <math>R</math> such that <math>J_1J_2\ldots J_n \subset I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>J_i \subset I</math>
			\|-
			\| 5 \|\| complement saturated subset version \|\| It is a [[proper ideal]] and its complement is a multiplicatively closed [[defining ingredient::saturated subset]] (that is, is closed with respect to the operations of multiplication and factorization). \|\| <math>I</math> is proper in <math>R</math> and the complement of <math>I</math> in <math>R</math> is a multiplicatively closed and saturated subset i.e. <math>ab \in R \setminus I \iff a \in R \setminus I</math> or <math>b \in R \setminus I</math>.
			\|-
			\| 6 \|\| quotient is integral domain \|\| The [[defining ingredient::quotient ring]] is an [[defining ingredient::integral domain]], i.e., a nonzero ring where the product of any two nonzero elements is nonzero. \|\| The quotient ring <math>R/I</math> is an integral domain.
			\|}

	===The set of prime ideals===		===The set of prime ideals===
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	The set of prime ideals in a commutative unital ring is termed its [[spectrum]]. The spectrum is more than just a set, it has the structure of a topological space. In fact, it is a locally ringed space.		The set of prime ideals in a commutative unital ring is termed its [[spectrum]]. The spectrum is more than just a set, it has the structure of a topological space. In fact, it is a locally ringed space.

			{{basicdef}}
			{{curing-ideal property}}
			{{quotient is a\|integral domain}}

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

Vipul at 20:25, 17 January 2009

2009-01-17T20:25:27Z

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

	* [[Maximal ideal]]		* [[Weaker than::Maximal ideal]]
	* [[Minimal prime ideal]]		* [[Weaker than::Minimal prime ideal]]
	* [[Principal prime ideal]]		* [[Weaker than::Principal prime ideal]]

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

	* [[Irreducible ideal]]		* [[Stronger than::Irreducible ideal]]
	* [[Primary ideal]]		* [[Stronger than::Primary ideal]]
	* [[Radical ideal]]		* [[Stronger than::Radical ideal]]
	* [[Ideal with prime radical]]		* [[Stronger than::Ideal with prime radical]]

	==Metaproperties==		==Metaproperties==

Vipul: 35 revisions

2008-05-12T16:33:13Z

35 revisions

← Older revision	Revision as of 16:33, 12 May 2008
(No difference)

Vipul: /* Symbol-free definition */

2008-03-03T19:12:45Z

Symbol-free definition

← Older revision		Revision as of 19:12, 3 March 2008
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	===Symbol-free definition===		===Symbol-free definition===

	An [[ideal]] in a [[commutative unital ring]] is termed a '''prime ideal''' if it is proper, and satisfies the following equivalent conditions:		An [[ideal]] in a [[commutative unital ring]] is termed a '''prime ideal''' if it is [[proper ideal\|proper]], and satisfies the following equivalent conditions:

	* Whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal		* Whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal

Vipul: /* Definition */

2008-03-03T17:25:30Z

Definition

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	* The complement of <math>I</math> in <math>R</math> is a multiplicatively closed and saturated subset i.e. <math>ab \in R \setminus I \iff a \in R \setminus I</math> or <math>b \in R \setminus I</math>		* The complement of <math>I</math> in <math>R</math> is a multiplicatively closed and saturated subset i.e. <math>ab \in R \setminus I \iff a \in R \setminus I</math> or <math>b \in R \setminus I</math>
	* The quotient ring <math>R/I</math> is an [[integral domain]]		* The quotient ring <math>R/I</math> is an [[integral domain]]

			===The set of prime ideals===

			{{further\|[[Spectrum of a commutative unital ring]]}}

			The set of prime ideals in a commutative unital ring is termed its [[spectrum]]. The spectrum is more than just a set, it has the structure of a topological space. In fact, it is a locally ringed space.

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

Vipul: /* External links */

2008-03-03T17:22:30Z

External links

← Older revision		Revision as of 17:22, 3 March 2008
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	\|}		\|}

			==References==
			* {{booklink\|Eisenbud}}, Page 12
	==External links==		==External links==
			{{searchbox\|"prime+ideal"}}
	===Definition links===		===Definition links===

Vipul: /* Definition with symbols */

2008-03-03T17:20:27Z

Definition with symbols

← Older revision		Revision as of 17:20, 3 March 2008
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	* Whenever <math>a, b \in R</math> are such that <math>ab \in I</math> then either <math>a \in I</math> or <math>b \in I</math>		* Whenever <math>a, b \in R</math> are such that <math>ab \in I</math> then either <math>a \in I</math> or <math>b \in I</math>
	* Whenever <math>J</math> and <math>K</math> are ideals such that <math>JK \subset I</math>, then either <math>J \subset I</math> or <math>K \subset I</math>		* Whenever <math>J</math> and <math>K</math> are ideals such that <math>JK \subset I</math>, then either <math>J \subset I</math> or <math>K \subset I</math>
			* Whenever <math>a_1,a_2,\ldots,a_n \in R</math> such that <math>a_1a_2 \ldots a_n \in I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>a_i \in I</math>
			* Whenever <math>J_1,J_2,\ldots,J_n</math> are [[ideal]]s in <math>R</math> such that <math>J_1J_2\ldots J_n \subset I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>J_i \subset I</math>
	* The complement of <math>I</math> in <math>R</math> is a multiplicatively closed and saturated subset i.e. <math>ab \in R \setminus I \iff a \in R \setminus I</math> or <math>b \in R \setminus I</math>		* The complement of <math>I</math> in <math>R</math> is a multiplicatively closed and saturated subset i.e. <math>ab \in R \setminus I \iff a \in R \setminus I</math> or <math>b \in R \setminus I</math>
	* The quotient ring <math>R/I</math> is an [[integral domain]]		* The quotient ring <math>R/I</math> is an [[integral domain]]

Vipul: /* Symbol-free definition */

2008-03-03T17:19:02Z

Symbol-free definition

← Older revision		Revision as of 17:19, 3 March 2008
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	* Whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal		* Whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal
	* Whenever the product of two ideals is contained in it, one of the ideals is contained in it		* Whenever the [[product of ideals\|product]] of two ideals is contained in it, one of the ideals is contained in it
			* Whenever the product of finitely many elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal
			* Whenever the product of finitely many ideals is contained in it, one of the ideals is contained in it
	* It is an ideal whose complement is a [[saturated subset]] (that is, is closed with respect to the operations of multiplication and factorization).		* It is an ideal whose complement is a [[saturated subset]] (that is, is closed with respect to the operations of multiplication and factorization).
	* The [[quotient ring]] by that ideal is an [[integral domain]]		* The [[quotient ring]] by that ideal is an [[integral domain]]

Vipul: /* In particular kinds of rings */

2008-03-03T17:15:02Z

In particular kinds of rings

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	{{further\|[[Prime ideal in affine ring]]}}		{{further\|[[Prime ideal in affine ring]]}}

			{\| class="wikitable" border="1"
			! Ring property assumed !! Nature of prime ideals
			\|-
			\|[[Zero-dimensional ring]] \|\| [[Maximal ideal\|maximal]]
			\|-
			\|[[One-dimensional domain]] \|\| Zero or [[maximal ideal\|maximal]]
			\|-
			\|[[Principal ideal domain]]\|\| Zero or [[maximal ideal\|maximal]]
			\|-
			\|[[Dedekind domain]]\|\|Zero or [[maximal ideal\|maximal]]
			\|}

	==External links==		==External links==