Krull dimension

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.