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[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] 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
ak{p}0\subsetneqak{p}1\subsetneq\ldots\subsetneqak{p}n
R
R
Given a prime ideal
ak{p}
ak{p}
\operatorname{ht}(ak{p})
ak{p}
ak{p}0\subsetneqak{p}1\subsetneq\ldots\subsetneqak{p}n=ak{p}
ak{p}
ak{p}
In a Noetherian ring, every prime ideal has finite height. Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension.[2] A ring is called catenary if any inclusion
ak{p}\subsetak{q}
ak{p}
ak{q}
ak{p}
ak{q}
In a Noetherian ring, a prime ideal has height at most n if and only if it is a minimal prime ideal over an ideal generated by 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.
More generally, the height of an ideal I is the infimum of the heights of all prime ideals containing I. In the language of algebraic geometry, this is the codimension of the subvariety of Spec(
R
It follows readily from the definition of the spectrum of a ring Spec(R), the space of prime ideals of R equipped with the Zariski topology, that the Krull dimension of R is equal to the dimension of its spectrum as a topological space, meaning the supremum of the lengths of all chains of irreducible closed subsets. This follows immediately from the Galois connection between ideals of R and closed subsets of Spec(R) and the observation that, by the definition of Spec(R), each prime ideal
ak{p}
ak{p}
ak{p}=(y2-x,y)\subsetC[x,y]
(0)=ak{p}0\subsetneq(y2-x)=ak{p}1\subsetneq(y2-x,y)=ak{p}2=ak{p}
f\inC[x,y,z]
I=(f3)
f ⋅ f2\inI
I
(f)
-infty
-1
\operatorname{gr}I(R)=
| infty | |
| oplus | |
| k=0 |
Ik/Ik+1
\operatorname{dim}\operatorname{gr}I(R)
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:
\dimRM:=\dim(R/{\operatorname{Ann}R(M)})
where AnnR(M), the annihilator, is the kernel of the natural map R → EndR(M) of R into the ring of R-linear endomorphisms of M.
In the language of schemes, finitely generated modules are interpreted as coherent sheaves, or generalized finite rank vector bundles.
The Krull dimension of a module over a possibly non-commutative ring is defined as the deviation of the poset of submodules ordered by inclusion. For commutative Noetherian rings, this is the same as the definition using chains of prime ideals.[6] The two definitions can be different for commutative rings which are not Noetherian.