Krull dimension

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In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals.

[edit] Explanation

If P₀, P₁, ... , P_n 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

$(2) \subsetneq (2,x) \subsetneq (2,x,y) \subsetneq (2,x,y,z)$

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 largest height of any prime ideal of R.

An integral domain of dimension zero is a field. Dedekind domains and discrete valuation rings have dimension one.

If a ring R has Krull dimension k, then the polynomial ring R[x] will have dimension between k + 1 and 2k + 1. If R is Noetherian, then the dimension of R[x] will be exactly 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.