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Krull dimension (Definition)

If $ R$ is a ring, the Krull dimension (or simply dimension) of $ R$, $ \dim R$ is the supremum of all integers $ n$ such that there is an increasing sequence of prime ideals $ \mathfrak{p}_0 \subsetneq \cdots \subsetneq \mathfrak{p}_n$ of length $ n$ in $ R$.

If $ X$ is a topological space, the Krull dimension (or simply dimension) of $ X$, $ \dim X$ is the supremum of all integers $ n$ such that there is a decreasing sequence of irreducible closed subsets $ F_0 \supsetneq \cdots \supsetneq F_n$ of $ X$.



"Krull dimension" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: height of a prime ideal, dimension

Other names:  dimension (Krull)

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bound on the Krull dimension of polynomial rings (Theorem) by mathcam
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Cross-references: closed subsets, irreducible, decreasing, topological space, length, prime ideals, sequence, increasing, integers, supremum, dimension, ring
There are 9 references to this entry.

This is version 4 of Krull dimension, born on 2001-12-20, modified 2006-10-25.
Object id is 1107, canonical name is KrullDimension.
Accessed 5430 times total.

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AMS MSC: 54-00 (General topology :: General reference works )
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