Krull dimension

In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899–1971), is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring.

A field k has Krull dimension 0; more generally, $k[x_{1},...,x_{n}]$ has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1.

Explanation

We say that a strict chain of inclusions of prime ideals of the form: ${\mathfrak {p}}_{0}\subsetneq {\mathfrak {p}}_{1}\subsetneq \ldots \subsetneq {\mathfrak {p}}_{n}$ is of length n. That is, it is counting the number of strict inclusions, not the number of primes, although these only differ by 1. Given a prime ${\mathfrak {p}}\subset R$, we define the height of ${\mathfrak {p}}$, written $\operatorname {ht} ({\mathfrak {p}})$ to be the supremum of the set $\{n\in \mathbb {N} :{\mathfrak {p}}{\text{ is the supremum of a strict chain of length }}n\}$

We define the Krull dimension of $R$ to be the supremum of the heights of all of its primes.

Nagata gave an example of a ring that has infinite Krull dimension even though every prime ideal has finite height. Nagata also gave an example of a Noetherian ring where not every chain can be extended to a maximal chain. Rings in which every chain of prime ideals can be extended to a maximal chain are known as catenary rings.

Krull dimension and schemes

It follows readily from the definition of the spectrum of a ring $\operatorname {Spec} (R)$, the space of prime ideals of $R$ equipped with the Zariski topology, that the Krull dimension of $R$ is precisely equal to the irreducible dimension of its spectrum. This follows immediately from the Galois connection between ideals of $R$ and closed subsets of $\operatorname {Spec} (R)$ and the observation that, by the definition of $\operatorname {Spec} (R)$, each prime ideal ${\mathfrak {p}}$ of $R$ corresponds to a generic point of the closed subset associated to ${\mathfrak {p}}$ via the Galois connection.

Examples

• The dimension of a polynomial ring over a field $k[x_{1},\ldots ,x_{d}]$ is the number of indeterminates d. These rings correspond to affine spaces in the language of schemes, so this result can be thought of as foundational. In general, if R is a Noetherian ring of dimension d, then the dimension of R[x] is d + 1. If the Noetherianity hypothesis is dropped, then R[x] can have dimension anywhere between d + 1 and 2d + 1.

• The ring of integers $\mathbb {Z}$ has dimension 1.

• An integral domain is a field if and only if its Krull dimension is zero. Dedekind domains that are not fields (for example, discrete valuation rings) have dimension one. In general, a Noetherian ring is Artinian if and only if its Krull dimension is 0.

Krull Dimension of a Module

If R is a commutative ring, and M is an R-module, we define the Krull dimension of M to be the Krull dimension of the quotient of R making M a faithful module. That is, we define it by the formula:

$\operatorname {dim} _{R}M:=\operatorname {dim} (R/\operatorname {Ann} _{R}(M))$

where $\operatorname {Ann} _{R}(M)$ , the annihilator, is the kernel of the natural map $R\to End_{R}(M)$ of R into the ring of $R$-linear endomorphisms on $M$.

In the language of schemes, finite type modules are interpreted as coherent sheaves, or generalized finite rank vector bundles.

See also

• Dimension theory (algebra)
• Regular local ring

Bibliography

• Irving Kaplansky, Commutative rings (revised ed.), University of Chicago Press, 1974, ISBN 0-226-42454-5. Page 32.
• A.I. Kostrikin and I.R. Shafarevich (edd), Algebra II, Encyclopaedia of Mathematical Scieinces 18, Springer-Verlag, 1991, ISBN 3-540-18177-6. Sect.4.7.