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.
Read more about this topic: Krull Dimension
Famous quotes containing the word explanation:
“Are cans constitutionally iffy? Whenever, that is, we say that we can do something, or could do something, or could have done something, is there an if in the offing—suppressed, it may be, but due nevertheless to appear when we set out our sentence in full or when we give an explanation of its meaning?”
—J.L. (John Langshaw)
“How strange a scene is this in which we are such shifting figures, pictures, shadows. The mystery of our existence—I have no faith in any attempted explanation of it. It is all a dark, unfathomed profound.”
—Rutherford Birchard Hayes (1822–1893)
“What causes adolescents to rebel is not the assertion of authority but the arbitrary use of power, with little explanation of the rules and no involvement in decision-making. . . . Involving the adolescent in decisions doesn’t mean that you are giving up your authority. It means acknowledging that the teenager is growing up and has the right to participate in decisions that affect his or her life.”
—Laurence Steinberg (20th century)