In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal I in a polynomial ring R is the Krull dimension of R/I.

A field k has Krull dimension 0; more generally, k[x₁, ..., x_n] has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent.

There are several other ways that have been used to define the dimension of a ring. Most of them coincide with the Krull dimension for Noetherian rings, but can differ for non-Noetherian rings.

We say that a chain of prime ideals of the form ${\displaystyle {\mathfrak {p}}_{0}\subsetneq {\mathfrak {p}}_{1}\subsetneq \ldots \subsetneq {\mathfrak {p}}_{n}}$ has length ${\displaystyle n}$. That is, the length is the number of strict inclusions, not the number of primes; these differ by ${\displaystyle 1}$. We define the Krull dimension of ${\displaystyle R}$ to be the supremum of the lengths of all chains of prime ideals in ${\displaystyle R}$.

Given a prime ideal ${\displaystyle {\mathfrak {p}}}$ in ${\displaystyle R}$, we define the height of ${\displaystyle {\mathfrak {p}}}$, written ${\displaystyle \operatorname {ht} ({\mathfrak {p}})}$, to be the supremum of the lengths of all chains of prime ideals contained in ${\displaystyle {\mathfrak {p}}}$, meaning that ${\displaystyle {\mathfrak {p}}_{0}\subsetneq {\mathfrak {p}}_{1}\subsetneq \ldots \subsetneq {\mathfrak {p}}_{n}={\mathfrak {p}}}$. In other words, the height of ${\displaystyle {\mathfrak {p}}}$ is the Krull dimension of the localization of ${\displaystyle R}$ at ${\displaystyle {\mathfrak {p}}}$. A prime ideal has height zero if and only if it is a minimal prime ideal. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals. The height is also sometimes called the codimension, rank, or altitude of a prime ideal.

In a Noetherian ring, every prime ideal has finite height. Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension. A ring is called catenary if any inclusion ${\displaystyle {\mathfrak {p}}\subset {\mathfrak {q}}}$ of prime ideals can be extended to a maximal chain of prime ideals between ${\displaystyle {\mathfrak {p}}}$ and ${\displaystyle {\mathfrak {q}}}$, and any two maximal chains between ${\displaystyle {\mathfrak {p}}}$ and ${\displaystyle {\mathfrak {q}}}$ have the same length. A ring is called universally catenary if any finitely generated algebra over it is catenary. Nagata gave an example of a Noetherian ring which is not catenary.

In a Noetherian ring, a prime ideal has height at most ${\displaystyle n}$ if and only if it is a minimal prime ideal over an ideal generated by ${\displaystyle n}$ elements (Krull's height theorem and its converse). It implies that the descending chain condition holds for prime ideals in such a way the lengths of the chains descending from a prime ideal are bounded by the number of generators of the prime.