Krull dimension

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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[x1,...,xn] has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1.



If P0, P1, ... , Pn are prime ideals of the ring such that P_0\subsetneq P_1\subsetneq \ldots \subsetneq P_n, 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. (Note here that since this ring is not an integral domain, the zero ideal is not prime.)

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. In particular, an integral domain has Krull dimension 1 when every nonzero prime ideal is maximal.

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. In general, a Noetherian ring is Artinian if and only if its Krull dimension is 0.

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.

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