Krull dimension: Difference between revisions

Revision as of 20:30, 20 January 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 exactly one. Any principal ideal domain, and more generally, any Dedekind domain, is one-dimensional.

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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.
+===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.