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height of a prime ideal (Definition)

Let $ R$ be a commutative ring and $ \mathfrak{p}$ a prime ideal of $ R$. The height of $ \mathfrak{p}$ is the supremum of all integers $ n$ such that there exists a chain

$\displaystyle \mathfrak{p}_0 \subset \cdots \subset \mathfrak{p}_n = \mathfrak{p}$
of distinct prime ideals. The height of $ \mathfrak{p}$ is denoted by $ \operatorname{h}(\mathfrak{p})$.

$ \operatorname{h}(\mathfrak{p})$ is also known as the rank of $ \mathfrak{p}$ and the codimension of $ \mathfrak{p}$.

The Krull dimension of $ R$ is the supremum of the heights of all the prime ideals of $ R$:

$\displaystyle \sup\lbrace \operatorname{h}(\mathfrak{p}) \mid \mathfrak{p}$ prime in $\displaystyle R \rbrace.$



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

Other names:  height
Also defines:  rank of an ideal, codimension of an ideal
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Cross-references: Krull dimension, codimension, rank, chain, integers, supremum, prime ideal, commutative ring
There are 7 references to this entry.

This is version 7 of height of a prime ideal, born on 2002-06-28, modified 2006-04-23.
Object id is 3146, canonical name is HeightOfAPrimeIdeal.
Accessed 4815 times total.

Classification:
AMS MSC: 14A99 (Algebraic geometry :: Foundations :: Miscellaneous)
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