Krull's principal ideal theorem - Revision history

Vipul: 8 revisions

2008-05-12T16:26:09Z

8 revisions

← Older revision	Revision as of 16:26, 12 May 2008
(No difference)

Vipul: /* Argument setup */

2008-01-20T21:35:42Z

Argument setup

← Older revision		Revision as of 21:35, 20 January 2008
Line 28:		Line 28:
	===Argument setup===		===Argument setup===

	Since <math>P</math> is minimal over <math>x</math>, we see that in the ring <math>R/(x)</math>, the ideal <math>P/(x)</math> is a maximal ideal ~~which~~ is also a minimal prime ideal. Thus, <math>R/(x)</math> is an [[Artinian ring]].		Since <math>P</math> is minimal over <math>x</math>, we see that in the ring <math>R/(x)</math>, the ideal <math>P/(x)</math> is the unique maximal ideal of a local ring, and is also a minimal prime ideal. Thus, <math>R/(x)</math> is a [[local Artinian ring]] with unique maximal ideal <math>P/(x)</math>.

	Hence, in <math>R</math>, consider the descending chain <math>x + Q^{(n)}</math>, where <math>Q^{(n)}</matH> denotes the <math>n^{th}</math> [[symbolic power]] of <math>Q</math>. This descending chain stabilizes, so we get:		Hence, in <math>R</math>, consider the descending chain <math>x + Q^{(n)}</math>, where <math>Q^{(n)}</matH> denotes the <math>n^{th}</math> [[symbolic power]] of <math>Q</math>. This descending chain stabilizes, so we get:

Vipul: /* Nakayama's lemma */

2008-01-20T21:34:06Z

Nakayama's lemma

← Older revision		Revision as of 21:34, 20 January 2008
Line 42:		Line 42:
	===Nakayama's lemma===		===Nakayama's lemma===

	{{~~fillin~~}}		The above reasoning shows that:

			<math>Q^{(n)} = (x)Q^{(n)} + Q^{(n+1)}</math>

			Now consider the module <math>M = Q^{(n)}/Q^{(n+1)}</math>. The above equation yields that:

			<math>M = (x)M</math>

			But since <math>x \in P</math>, we see that <math>x</math> is in the Jacobson radical of <math>R</math>, so [[Nakayama's lemma]] yields that <math>M = 0</math>. Thus <math>Q^{(n)} = Q^{(n+1)}</math>.

			We now apply Nakayama's lemma in the [[localization at a prime ideal\|localization]] at <math>Q</math>, to conclude that:

			<math>(Q_Q)^n = 0</math>

			This yields that <math>R_Q</math> isa [[zero-dimensional ring]].

Vipul at 21:28, 20 January 2008

2008-01-20T21:28:48Z

@@ Line 1: / Line 1: @@
 {{curing property implication}}
 ==Statement==
@@ Line 15: / Line 17: @@
 * [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem.
 * [[Determinantal ideal theorem]]: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix

Vipul: /* Statement */

2008-01-20T21:10:59Z

Statement

← Older revision		Revision as of 21:10, 20 January 2008
Line 5:		Line 5:
	===Symbolic statement===		===Symbolic statement===

	Let <math>R</math> be a [[Noetherian ring\|Noetherian]] and <math>x \in R</math>. Let <math>P</math> be a minimal prime ideal ~~among those~~ containing <math>x</math>. Then, the codimension of <math>P</math> is at most 1.		Let <math>R</math> be a [[Noetherian ring\|Noetherian]] and <math>x \in R</math>. Let <math>P</math> be a minimal element among [[prime ideal]]s containing <math>x</math>. Then, the [[codimension]] of <math>P</math> is at most 1.

	===Property-theoretic statement===		===Property-theoretic statement===

	The [[property of commutative unital rings]] of being a [[Noetherian ring]] is stronger than the property of being a [[ring satisfying PIT]].		The [[property of commutative unital rings]] of being a [[Noetherian ring]] is stronger than the property of being a [[ring satisfying PIT]].

	==Generalizations==		==Generalizations==

	* [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem.		* [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem.
	* [[Determinantal ideal theorem]]: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix		* [[Determinantal ideal theorem]]: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix

Vipul at 23:43, 7 January 2008

2008-01-07T23:43:43Z

← Older revision		Revision as of 23:43, 7 January 2008
Line 1:		Line 1:
	{{~~result for ring-type\|Noetherian ring~~}}		{{curing property implication}}

	==Statement==		==Statement==

			===Symbolic statement===

	Let <math>R</math> be a [[Noetherian ring\|Noetherian]] and <math>x \in R</math>. Let <math>P</math> be a minimal prime ideal among those containing <math>x</math>. Then, the codimension of <math>P</math> is at most 1.		Let <math>R</math> be a [[Noetherian ring\|Noetherian]] and <math>x \in R</math>. Let <math>P</math> be a minimal prime ideal among those containing <math>x</math>. Then, the codimension of <math>P</math> is at most 1.

			===Property-theoretic statement===

			The [[property of commutative unital rings]] of being a [[Noetherian ring]] is stronger than the property of being a [[ring satisfying PIT]].
	==Generalizations==		==Generalizations==

	* [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem.		* [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem.
	* [[Determinantal ideal theorem]]: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix		* [[Determinantal ideal theorem]]: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix

Vipul: /* Generalizations */

2007-08-10T08:32:31Z

Generalizations

← Older revision		Revision as of 08:32, 10 August 2007
Line 8:		Line 8:

	* [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem.		* [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem.
			* [[Determinantal ideal theorem]]: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix

Vipul: /* Statement */

2007-08-07T10:00:04Z

Statement

← Older revision		Revision as of 10:00, 7 August 2007
Line 3:		Line 3:
	==Statement==		==Statement==

	Let <math>R</math> be a [[Noetherian ring\|Noetherian]] and <math>x \in R</math>. Let <math>P</~~mat~~> be a minimal prime ideal among those containing <math>x</math>. Then, the codimension of <math>P</math> is at most 1.		Let <math>R</math> be a [[Noetherian ring\|Noetherian]] and <math>x \in R</math>. Let <math>P</math> be a minimal prime ideal among those containing <math>x</math>. Then, the codimension of <math>P</math> is at most 1.

	==Generalizations==		==Generalizations==

	* [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem.		* [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem.

Vipul at 09:59, 7 August 2007

2007-08-07T09:59:23Z

New page

{{result for ring-type|Noetherian ring}}

==Statement==

Let <math>R</math> be a [[Noetherian ring|Noetherian]] and <math>x \in R</math>. Let <math>P</mat> be a minimal prime ideal among those containing <math>x</math>. Then, the codimension of <math>P</math> is at most 1.

==Generalizations==

* [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem.