# Krull dimension

From Commalg

Template:Curing-dimension notion

## Contents

## 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 be a commutative unital ring. The Krull dimension of , denoted is the supremum over all for which there exist strictly descending chains of prime ideals:

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