In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is the number of strict inclusions in a maximal 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[x1,...,xn] has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1.
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Explanation
If P0, P1, ... , Pn are prime ideals of the ring such that 
, then these prime ideals form a chain of length n. The Krull dimension is the supremum of the lengths of chains of prime ideals.
For example, in the ring (Z/8Z)[x,y,z] we can consider the chain
Each of these ideals is prime, so the Krull dimension of (Z/8Z)[x, y, z] is at least 3. In fact the dimension of this ring is exactly 3.
An alternate way of phrasing this definition is to say that the Krull dimension of R is the supremum of heights of all prime ideals of R. Nagata gave an example of a ring that has infinite Krull dimension even though every prime ideal has finite height.[citation needed] Nagata also gave an example of a Noetherian ring where not every chain can be extended to a maximal chain.[1] Rings in which every chain of prime ideals can be extended to a maximal chain are known as catenary rings.
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.
If a ring R has Krull dimension k, then the polynomial ring R[x] will have dimension at least k + 1 and at most 2k + 1. If R is Noetherian, then the dimension of R[x] is k + 1.
If K is a field and R is a finitely generated K-algebra, then R can be identified with the ring of polynomial functions on an affine variety X defined over K and the Krull dimension of R equals the usual dimension of the variety X.
See also
References
- ^ Nagata, M. Local Rings (1962). Wiley, New York.
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.
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