Height of truncated system of parameter

Question

Let $R$ be a Noetherian local ring of dimension $d$, and $a_1,\dots,a_d$ is a system of parameters. I am wondering whether the following statement is true:

$\mathrm{ht}(a_1,\dots,a_i)=i$ for all $i$, $1 \le i \le d$.

I am thinking about this because by definition, $\mathrm{ht}(a_1,\dots,a_d)=d$ thus the statement holds for $i=d$. And if $R$ is Cohen-Macaulay, then $a_1,\dots,a_i$ is a regular sequence, thus $\dim R/(a_1,\dots,a_i)=d-i$ and $\dim R/I + \operatorname{ht} I = \dim R$ holds for any ideal in a CM local ring, thus the statement holds for any $I$. So I am curious about whether counter example exists in a non CM noetherian local ring. (Or even local Noetherian domain.)

Answer 1

This is true if $R$ is catenary and equidimensional. However, for example if $R$ is not equidimensional, then there exists a minimal prime $\mathfrak{p}$ of $R$ such that dim$(R/\mathfrak{p})<\text{dim}(R)$. Therefore any $x\in \mathfrak{p}$ which not in the union of the other minimal primes is a parameter element of height $0$.