Krull dimension - Revision history

Vipul: 8 revisions

2008-05-12T16:26:12Z

8 revisions

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

Vipul: /* Facts */

2008-03-17T00:26:50Z

Facts

← Older revision		Revision as of 00:26, 17 March 2008
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	==Facts==		==Facts==

	The [[polynomial ring]] over any [[Noetherian ring]] of finite dimension, has dimension one more than the original ring. Thus, we see that the polynomial ring in <math>n</math> variables over a field, has dimension <math>n</math>, while the polynomial ring in <math>n</math> variables over a principal ideal domain (or Dedekind domain) which is not a field, has dimension <math>n + 1</math>		* The [[polynomial ring]] over any [[Noetherian ring]] of finite dimension, has dimension one more than the original ring. Thus, we see that the polynomial ring in <math>n</math> variables over a field, has dimension <math>n</math>, while the polynomial ring in <math>n</math> variables over a principal ideal domain (or Dedekind domain) which is not a field, has dimension <math>n + 1</math>
			* For a [[Noetherian local ring]], the Krull dimension equals the degree of its [[Hilbert-Samuel polynomial]].

Vipul: /* Related ring properties */

2008-01-20T22:05:03Z

Related ring properties

← Older revision		Revision as of 22:05, 20 January 2008
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	* [[Zero-dimensional ring]] is a ring whose Krull dimension is zero. Particular examples of such rings are [[Artinian ring]]s and completely local rings.		* [[Zero-dimensional ring]] is a ring whose Krull dimension is zero. Particular examples of such rings are [[Artinian ring]]s and completely local rings.
	* Any [[integral domain]] which is not a [[field]] must have dimension at least one. A [[one-dimensional domain]] is an integral domain which has Krull dimension ~~exactly~~ one. Any [[principal ideal domain]], and more generally, any [[Dedekind domain]], is one-dimensional.		* Any [[integral domain]] which is not a [[field]] must have dimension at least one. A [[one-dimensional domain]] is an integral domain which has Krull dimension at most one. Any [[principal ideal domain]], and more generally, any [[Dedekind domain]], is one-dimensional.
	* A [[finite-dimensional ring]] is a ring with finite Krull dimension; a [[finite-dimensional domain]] is an integral domain with finite Krull dimension.		* A [[finite-dimensional ring]] is a ring with finite Krull dimension; a [[finite-dimensional domain]] is an integral domain with finite Krull dimension.

Vipul at 22:04, 20 January 2008

2008-01-20T22:04:47Z

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	* [[Zero-dimensional ring]] is a ring whose Krull dimension is zero. Particular examples of such rings are [[Artinian ring]]s and completely local rings.		* [[Zero-dimensional ring]] is a ring whose Krull dimension is zero. Particular examples of such rings are [[Artinian ring]]s and completely local rings.
	* Any [[integral domain]] which is not a [[field]] must have dimension at least one. A [[one-dimensional domain]] is an integral domain which has Krull dimension exactly one. Any [[principal ideal domain]], and more generally, any [[Dedekind domain]], is one-dimensional.		* Any [[integral domain]] which is not a [[field]] must have dimension at least one. A [[one-dimensional domain]] is an integral domain which has Krull dimension exactly one. Any [[principal ideal domain]], and more generally, any [[Dedekind domain]], is one-dimensional.
			* A [[finite-dimensional ring]] is a ring with finite Krull dimension; a [[finite-dimensional domain]] is an integral domain with finite Krull dimension.

			==Facts==

			The [[polynomial ring]] over any [[Noetherian ring]] of finite dimension, has dimension one more than the original ring. Thus, we see that the polynomial ring in <math>n</math> variables over a field, has dimension <math>n</math>, while the polynomial ring in <math>n</math> variables over a principal ideal domain (or Dedekind domain) which is not a field, has dimension <math>n + 1</math>

Vipul at 20:30, 20 January 2008

2008-01-20T20:30:12Z

← Older revision		Revision as of 20:30, 20 January 2008
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	The '''Krull dimension''' of a [[commutative unital ring]] is the supremum of lengths of descending chains of distinct [[prime ideal]]s.		The '''Krull dimension''' of a [[commutative unital ring]] is the supremum of lengths of descending chains of distinct [[prime ideal]]s.

			===Definition with symbols===

			Let <math>R</math> be a [[commutative unital ring]]. The Krull dimension of <math>R</math>, denoted <math>dim(R)</math> is the supremum over all <math>n</math> for which there exist strictly descending chains of [[prime ideal]]s:

			<math>P_0 \supset P_1 \supset \ldots \supset P_n</math>

			==Related ring properties==

			* [[Zero-dimensional ring]] is a ring whose Krull dimension is zero. Particular examples of such rings are [[Artinian ring]]s and completely local rings.
			* Any [[integral domain]] which is not a [[field]] must have dimension at least one. A [[one-dimensional domain]] is an integral domain which has Krull dimension exactly one. Any [[principal ideal domain]], and more generally, any [[Dedekind domain]], is one-dimensional.

Vipul at 21:13, 5 January 2008

2008-01-05T21:13:34Z

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	{{dimension notion}}		{{curing-dimension notion}}

	==Definition==		==Definition==

Vipul at 21:12, 5 January 2008

2008-01-05T21:12:45Z

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	{{~~commring~~ dimension notion}}		{{dimension notion}}

	==Definition==		==Definition==

Vipul at 16:20, 30 June 2007

2007-06-30T16:20:50Z

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			{{commring dimension notion}}

	==Definition==		==Definition==

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	The '''Krull dimension''' of a [[commutative unital ring]] is the supremum of lengths of descending chains of distinct [[prime ideal]]s.		The '''Krull dimension''' of a [[commutative unital ring]] is the supremum of lengths of descending chains of distinct [[prime ideal]]s.

	~~[[Category: Numerical invariants of commutative rings]]~~
	~~[[Category: Notions of dimension]]~~

Vipul: Started the page

2007-01-09T07:28:35Z

Started the page

New page

==Definition==

===Symbol-free definition===

The '''Krull dimension''' of a [[commutative unital ring]] is the supremum of lengths of descending chains of distinct [[prime ideal]]s.

[[Category: Numerical invariants of commutative rings]]
[[Category: Notions of dimension]]