Krull dimension

From Academic Kids

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. We take the supremum of chain lengths if no maximal chain can be found. For example, in the ring (Z/8Z)[x,y,z] we can consider the chain

(2) ⊂ (2,x) ⊂ (2,x,y) ⊂ (2,x,y,z)

Each of these ideals is prime, so the Krull dimension of (Z/8Z)[x, y, z] is at least the number of strict inclusions in this chain, that is, 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.

According to this convention, a 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.

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