Corollary 1

a) Every noetherian local ring is finite-dimensional. More generally, every nonzero noetherian semi-local ring (II, § 3, No. 5, def. 4) is finite-dimensional. b) Let $A$ be a noetherian ring. Every ideal of $A$, distinct from $A$, is of finite height. c) Every decreasing sequence of prime ideals of a noetherian ring $A$ is stationary.

a) Let $A$ be a nonzero semi-local noetherian ring and $a$ its radical; the quotient ring $A/a$ is artinian and nonzero, hence of dimension 0 ($§ 1$, No. 3, Example 1). There exists an integer $m \geqslant 0$ such that the ideal $a$ of $A$ is generated by $m$ elements; hence $0 \leqslant \dim(A) \leqslant m$ by Prop. 2. b) Let $a \neq A$ be an ideal of $A$, and let $m$ be a maximal ideal of $A$ containing $a$. We have $0 \leqslant \mathrm{ht}(a) \leqslant \dim(A_m)$ by Prop. 7 of $§ 1$, No. 3, and $A_m$ is a local noetherian ring. Hence $\mathrm{ht}(a)$ is finite by a). c) Every finite strictly decreasing sequence $(p_i)_{0 \leqslant i \leqslant n}$ of prime ideals of $A$ defines a chain $p_n \subset ... \subset p_0$, whence $n \leqslant \dim(A_{p_0}) < + \infty$. Therefore there cannot exist an infinite strictly decreasing sequence of prime ideals of $A$, whence c).