High Quality Content by WIKIPEDIA articles! 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. 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.

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