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, ..., xn] 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.
the Krulldimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension...
local rings at points of V have the same Krulldimension (see ); thus: If V is a variety, the Krulldimension of the local ring at any point of V This...
the dimension as vector space is finite if and only if its Krulldimension is 0. For any normal topological space X, the Lebesgue covering dimension of...
finite global dimension and in that case the global dimension is the Krulldimension of R. The significance of this is that a global dimension is a homological...
polynomial with integer coefficients, and δ {\displaystyle \delta } is the Krulldimension of S. In this case the series expansion of this rational fraction is...
field k, the Krulldimension of its coordinate ring equals the transcendence degree of its function field, and this defines the dimension of X. It follows...
polynomial ring S = k [y1, y2, ..., yd]. The integer d is equal to the Krulldimension of the ring A; and if A is an integral domain, d is also the transcendence...
local rings those rings which are regular. Their global dimension coincides with the Krulldimension, whose definition is module-theoretic. When the ring...
Noetherian ring. Artinian Ascending chain condition for principal ideals Krulldimension Maximal condition on congruences Noetherian Proof: first, a strictly...
of the Krulldimension of a ring, first for Noetherian rings before moving on to expand his theory to cover general valuation rings and Krull rings. To...
concerning commutative rings stem from geometric intuition. The Krulldimension (or dimension) dim R of a ring R measures the "size" of a ring by, roughly...
curve; i.e., the local ring R at the point is a regular local ring of Krulldimension one or a discrete valuation ring. For example, consider the inclusion...
Regular local ring Discrete valuation ring Global dimension Regular sequence (algebra) KrulldimensionKrull's principal ideal theorem Primary ideal Primary...
commutative ring to its internal ring structure, particularly its Krulldimension and depth. The following list given by Melvin Hochster is considered...
In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix...
sequence. Krull ring A Krull ring (or Krull domain) is a ring with a well behaved theory of prime factorization. Krulldimension See dimension. Laskerian...
on itself Regular ring, a ring such that all its localizations have Krulldimension equal to the minimal number of generators of the maximal ideal von...
not a field. R is an integrally closed Noetherian local ring with Krulldimension one. R is a principal ideal domain with a unique non-zero prime ideal...
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