Example 4

Let $A$ be a ring. One has

In fact, if $p_0 \subset ... \subset p_n$ is a chain of prime ideals of $A$, of length $n$, one obtains a chain $p'_0 \subset ... \subset p'_{n+1}$ of prime ideals of $A[X]$, of length $n + 1$, by putting $p'_i = p_i A[X]$ for $0 \leq i \leq n$, and $p'_{n+1} = p_n A[X] + XA[X]$.

By the same reasoning, one proves the inequality $\dim(A[[X]]) \geqslant \dim(A) + 1$. One deduces by induction the inequalities

We shall prove later (§ 3, No. 4, cor. 3 to prop. 7 and cor. 3 to prop. 8) that one has equality in the two preceding formulae when $A$ is noetherian.