Primary ideal - Revision history

Vipul: /* Relation with other properties */

2011-12-18T17:56:01Z

Relation with other properties

← Older revision		Revision as of 17:56, 18 December 2011
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	{{quotient is a\|primary ring}}		{{quotient is a\|primary ring}}

			==Examples==

			===Important ring types===

			{\| class="sortable" border="1"
			! Property of commutative unital rings !! Characterization of primary ideals in such a ring !! Examples
			\|-
			\| [[principal ideal domain]] (i.e., an [[integral domain]] in which every ideal is a [[principal ideal]]) \|\| either the zero ideal or a principal ideal generated by a nontrivial (i.e., use nonzero exponent) power of a prime element . In particular, for a PID, the primary ideals coincide with the powers of prime ideals, and also with the [[ideal with prime radical\|ideals with prime radical]]. \|\| In the [[ring of integers]], the primary ideals are precisely the zero ideal and the ideals generated by powers of primes. Thus, <math>3\mathbb{Z}, 4 \mathbb{Z}, 5\mathbb{Z}, 8\mathbb{Z}</math> are primary ideals. On the other hand, <math>6\mathbb{Z}</math> and <math>24\mathbb{Z}</math> are not primary.<br>In the (univariate) [[polynomial ring over a field]], the primary ideals are precisely the principal ideals generated by a power of an irreducible polynomial.
			\|-
			\| [[Dedekind domain]] (roughly speaking, unique factorization of ideals into prime ideals) \|\| coincides with [[power of a prime ideal]] \|\|
			\|-
			\| [[unique factorization domain]] \|\| the ''principal'' primary ideals are precisely the ideals generated by powers of primes. The non-principal primary ideals are harder to describe. \|\| In the [[multivariate polynomial ring over a field]], ...
			\|}
	==Relation with other properties==		==Relation with other properties==

Vipul: /* Definition */

2011-12-18T17:21:34Z

Definition

← Older revision		Revision as of 17:21, 18 December 2011
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	\| 3 \|\|quotient ring is primary \|\| the quotient ring is a [[primary ring]], i.e., it has exactly one associated prime. \|\| the quotient ring <math>R/I</math> is a primary ring, i.e., it has exactly one associated prime.		\| 3 \|\|quotient ring is primary \|\| the quotient ring is a [[primary ring]], i.e., it has exactly one associated prime. \|\| the quotient ring <math>R/I</math> is a primary ring, i.e., it has exactly one associated prime.
	\|}		\|}
			{{TOCright}}
	{{curing-ideal property}}		{{curing-ideal property}}
	{{quotient is a\|primary ring}}		{{quotient is a\|primary ring}}

Vipul at 17:21, 18 December 2011

2011-12-18T17:21:01Z

← Older revision		Revision as of 17:21, 18 December 2011
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	~~{{curing-ideal property}}~~

	~~{{quotient is a\|primary ring}}~~

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

	~~An [[ideal]]~~ in ~~a [[commutative unital ring]] is termed '''primary''' if it satisfies the following equivalent conditions:~~		===Equivalent definitions in tabular format===
	* Whenever the product of two elements of the ring lies inside the ideal, either the first element lies inside the ideal or a suitable power of the second element lies inside the ideal
	* There is exactly one [[associated prime to an ideal\|associated prime to the ideal]], i.e. exactly one [[associated prime to a module\|associated prime]] to the quotient ring

	==~~=Definition with symbols===~~		{\| class="sortable" border="1"
			! No. !! Shorthand !! An [[ideal]] in a [[commutative unital ring]] is termed a primary ideal if ... !! An [[ideal]] <math>I</math> in a [[commutative unital ring]] is termed a primary ideal if ...
			\|-
			\| 1 \|\| product of two elements version \|\| it is a [[proper ideal]] in the ring and whenever the product of two elements of the ring lies in the ideal, either the first element lies in the ideal or some power of the second element lies in the ideal. \|\| <math>I</math> is proper in <math>R</math> and whenever <math>a,b \in R</math> (possibly equal, possibly distinct) are such that <math>ab \in I</math>, then either <math>a \in I</math> or there exists a positive integer <math>n</math> such that <math>b^n \in I</math>.
			\|-
			\| 2 \|\| exactly one associated prime \|\| there is exactly one [[associated prime to an ideal\|associated prime to the ideal]] \|\| {{fillin}}
			\|-
			\| 3 \|\|quotient ring is primary \|\| the quotient ring is a [[primary ring]], i.e., it has exactly one associated prime. \|\| the quotient ring <math>R/I</math> is a primary ring, i.e., it has exactly one associated prime.
			\|}

	~~An [[~~ideal~~]] <math>I</math> in~~ a ~~commutative~~ ring <math>R</math> is termed '''primary''' if for any <math>p,q</math> in <math>R</math> such that <math>pq</math> is in <math>I</math>, either <math>p</math> is in <math>I</math>, or there exists a positive integer <math>n</math> such that <math>q^n</math> lies in <math>I</math>.		{{curing-ideal property}}
			{{quotient is a\|primary ring}}

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

Vipul: /* Relation with other properties */

2009-01-17T20:26:04Z

Relation with other properties

← Older revision		Revision as of 20:26, 17 January 2009
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	===Stronger properties===		===Stronger properties===

	* [[Maximal ideal]]		* [[Weaker than::Maximal ideal]]
	* [[Prime ideal]]		* [[Weaker than::Prime ideal]]
	* [[Ideal with maximal radical]]		* [[Weaker than::Ideal with maximal radical]]
	* [[Irreducible ideal]] if the ring is [[Noetherian ring\|Noetherian]]: {{proofat\|[[Irreducible implies primary (Noetherian)]]}}		* [[Irreducible ideal]] if the ring is [[Noetherian ring\|Noetherian]]: {{proofat\|[[Irreducible implies primary (Noetherian)]]}}

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

	* [[Ideal with prime radical]]		* [[Stronger than::Ideal with prime radical]]

	===Incomparable properties===		===Incomparable properties===

	* [[Radical ideal]]		* [[Radical ideal]]

Vipul: 6 revisions

2008-05-12T16:28:55Z

6 revisions

← Older revision	Revision as of 16:28, 12 May 2008
(No difference)

Vipul: /* Definition */

2008-03-02T03:24:42Z

Definition

← Older revision		Revision as of 03:24, 2 March 2008
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	===Symbol-free definition===		===Symbol-free definition===

	An [[ideal]] in a [[commutative unital ring]] ~~(or in any [[commutative ring]])~~ is termed '''primary''' if it satisfies the ~~condition that whenever~~ the product of two elements of the ring lies inside the ideal, either the first element lies inside the ideal or a suitable power of the second element lies inside the ideal.		An [[ideal]] in a [[commutative unital ring]] is termed '''primary''' if it satisfies the following equivalent conditions:
			* Whenever the product of two elements of the ring lies inside the ideal, either the first element lies inside the ideal or a suitable power of the second element lies inside the ideal
			* There is exactly one [[associated prime to an ideal\|associated prime to the ideal]], i.e. exactly one [[associated prime to a module\|associated prime]] to the quotient ring

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

Vipul: /* Stronger properties */

2008-01-10T01:07:07Z

Stronger properties

← Older revision		Revision as of 01:07, 10 January 2008
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	* [[Prime ideal]]		* [[Prime ideal]]
	* [[Ideal with maximal radical]]		* [[Ideal with maximal radical]]
	* [[Irreducible ideal]] if the ring is [[Noetherian ring\|Noetherian]]: {{proofat\|[[Irreducible implies primary (Noetherian)]]		* [[Irreducible ideal]] if the ring is [[Noetherian ring\|Noetherian]]: {{proofat\|[[Irreducible implies primary (Noetherian)]]}}

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

Vipul at 01:05, 10 January 2008

2008-01-10T01:05:39Z

← Older revision		Revision as of 01:05, 10 January 2008
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	{{~~commring~~-ideal property}}		{{curing-ideal property}}

	{{quotient is a\|primary ring}}		{{quotient is a\|primary ring}}

	==Definition ~~for commutative rings~~==		==Definition==

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

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	An [[ideal]] <math>I</math> in a commutative ring <math>R</math> is termed '''primary''' if for any <math>p,q</math> in <math>R</math> such that <math>pq</math> is in <math>I</math>, either <math>p</math> is in <math>I</math>, or there exists a positive integer <math>n</math> such that <math>q^n</math> lies in <math>I</math>.		An [[ideal]] <math>I</math> in a commutative ring <math>R</math> is termed '''primary''' if for any <math>p,q</math> in <math>R</math> such that <math>pq</math> is in <math>I</math>, either <math>p</math> is in <math>I</math>, or there exists a positive integer <math>n</math> such that <math>q^n</math> lies in <math>I</math>.

	==~~Definition for non-commutative rings~~==		==Relation with other properties==

			===Stronger properties===

			* [[Maximal ideal]]
			* [[Prime ideal]]
			* [[Ideal with maximal radical]]
			* [[Irreducible ideal]] if the ring is [[Noetherian ring\|Noetherian]]: {{proofat\|[[Irreducible implies primary (Noetherian)]]
			===Weaker properties===

			* [[Ideal with prime radical]]

			===Incomparable properties===

	~~The same definition applies verbatim for noncommutative rings. See~~ [[~~Primary~~ ideal ~~(noncommutative rings)~~]] ~~for more details.~~		* [[Radical ideal]]

Vipul at 09:36, 7 August 2007

2007-08-07T09:36:40Z

← Older revision		Revision as of 09:36, 7 August 2007
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	{{commring-ideal property}}		{{commring-ideal property}}

	{{~~quot-determined property~~\|[[primary ring]]}}		{{quotient is a\|primary ring}}

	==Definition for commutative rings==		==Definition for commutative rings==

Vipul at 16:02, 30 June 2007

2007-06-30T16:02:25Z

← Older revision		Revision as of 16:02, 30 June 2007
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			{{commring-ideal property}}

			{{quot-determined property\|[[primary ring]]}}

	==Definition for commutative rings==		==Definition for commutative rings==

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	The same definition applies verbatim for noncommutative rings. See [[Primary ideal (noncommutative rings)]] for more details.		The same definition applies verbatim for noncommutative rings. See [[Primary ideal (noncommutative rings)]] for more details.

	~~[[Category: Properties of ideals in commutative rings]]~~
	~~[[Category: Quotient-determined properties of ideals in commutative rings]]~~