ideals in .

Given a prime ideal in , we define the ' of , written , to be the supremum of the lengths of all chains of prime ideals contained in , meaning that . In other words, the height of is the Krull dimension of the localization of at . A prime ideal has height zero if and only if it is a minimal prime ideal. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals. The height