Krull dimension: Difference between revisions

Revision as of 00:26, 17 March 2008

Definition

Symbol-free definition

The Krull dimension of a commutative unital ring is the supremum of lengths of descending chains of distinct prime ideals.

Definition with symbols

Let $R$ be a commutative unital ring. The Krull dimension of $R$ , denoted $dim(R)$ is the supremum over all $n$ for which there exist strictly descending chains of prime ideals:

$P_{0}\supset P_{1}\supset \ldots \supset P_{n}$

Related ring properties

Zero-dimensional ring is a ring whose Krull dimension is zero. Particular examples of such rings are Artinian rings 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 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.

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 $n$ variables over a field, has dimension $n$ , while the polynomial ring in $n$ variables over a principal ideal domain (or Dedekind domain) which is not a field, has dimension $n+1$
For a Noetherian local ring, the Krull dimension equals the degree of its Hilbert-Samuel polynomial.

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 ==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]].