Explanation
We say that a strict chain of inclusions of prime ideals of the form: is of length n. That is, it is counting the number of strict inclusions, not the number of primes, although these only differ by 1. Given a prime, we define the height of, written to be the supremum of the set
We define the Krull dimension of to be the supremum of the heights of all of its primes.
Nagata gave an example of a ring that has infinite Krull dimension even though every prime ideal has finite height. Nagata also gave an example of a Noetherian ring where not every chain can be extended to a maximal chain. Rings in which every chain of prime ideals can be extended to a maximal chain are known as catenary rings.
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