Krull dimension

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"1, ..., "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.
Explanation.
We say that a chain of prime ideals of the form
formula_1
has length n. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define the Krull dimension of formula_2 to be the supremum of the lengths of all chains of prime ideals in formula_2.
Given a prime formula_4 in "R", we define the of formula_4, written formula_6, to be the supremum of the lengths of all chains of prime ideals contained in formula_4, meaning that formula_8. In other words, the height of formula_4 is the Krull dimension of the localization of "R" at formula_4. 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 formula_11 of prime ideals can be extended to a maximal chain of prime ideals between formula_4 and formula_13, and any two maximal chains between formula_4
and formula_13 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 "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(formula_2) corresponding to I.
Schemes.
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 formula_4 of "R" corresponds to a generic point of the closed subset associated to formula_4 by the Galois connection.
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:
where Ann"R"("M"), the annihilator, is the kernel of the natural map R → End"R"(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.
For non-commutative rings.
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. The two definitions can be different for commutative rings which are not Noetherian.