Theorem 3

Let $A$ be a $k$-algebra of finite type. a) For every minimal prime ideal $p$ of $A$, all maximal chains of prime ideals of $A$ with origin $p$ have length the integer $\deg.\mathrm{tr}_k \kappa(p)$. b) The ring $A$ is catenary and its dimension is the supremum of the integers $\deg.\mathrm{tr}_k \kappa(p)$, where $p$ runs through the minimal prime ideals of $A$. c) If $A$ is integral, then all maximal chains of prime ideals of $A$ have the same length, and the dimension of $A$ is the degree of transcendence over $k$ of the field of fractions of $A$.

Suppose $A$ is integral and consider a maximal chain $p_0 \subset ... \subset p_m$ of prime ideals of $A$. We have $p_0 = 0$. We deduce then from Lemma 3 the existence of an injective homomorphism $\varphi : k[X_1, ..., X_m] \to A$ of $k$-algebras which makes $A$ a finite $k[X_1, ..., X_m]$-algebra. Consequently, the degree of transcendence over $k$ of the field of fractions of $A$ is equal to $m$, whence c). Assertion a) follows from assertion c) applied to the ring $A/p$ and assertion b) is an immediate consequence of a) and Prop. 5 of § 1, No. 2.