Non-discrete valuation rings form a great class of examples.

**Remark.** Recall that if $(\Gamma,\geq)$ is a totally ordered abelian group, then we can obtain a valuation ring in the following manner. Let $M = \{m \in \Gamma\ \big|\ m \geq 0\}$ be the submonoid of nonnegative elements, and let $k[M]$ be the monoid algebra on $M$. Then we obtain a nonarchimedean valuation on $k[M]$ by
\begin{align*}
v \colon k[M] &\to M \cup \{\infty\}\\
\sum a_m m &\mapsto \min\left\{m\ \big|\ a_m \neq 0\right\},
\end{align*}
where the minimum is taken with respect to the ordering of $\Gamma$, and by convention $v(0) = \infty$.

The set $\mathfrak m = \{x \in k[M]\ \big|\ v(x) > 0\}$ is a maximal ideal, and the ring $R = k[M]_\mathfrak m$ is local domain with a valuation taking values in $M$. (Prime) ideals in $M$ correspond to (prime) ideals in $R$.

Moreover, $\mathfrak m$ is finitely generated if and only if the corresponding ideal $\{m \in M\ |\ m > 0\}$ in $M$ is finitely generated. We will denote the latter set by $\mathfrak m$ as well, as we will be working exclusively in ordered group language.

So for valuation rings, the question is equivalent to the following:

**Question.** Does there exist a totally ordered group $\Gamma$ such that $\mathfrak m$ is finitely generated, but $M$ has an infinite chain of prime ideals?

My example is motivated by, but not identical to, this example of a valuation ring of infinite Krull dimension (but I don't think the maximal ideal is finitely generated in that example).

**Example.** Let $\Gamma = \mathbb Z[x]$, with the total ordering given by
$$p \succeq q \Longleftrightarrow p(x) \geq q(x) \text{ as } x\to \infty.$$
It is a total ordering because $p-q$ is eventually positive or eventually negative for $p \neq q$. The explicit description for $p = \sum_{i = 0}^m a_i x^i$ and $q = \sum_{j = 0}^n b_j x^j$ is: we have $p \succeq q$ if and only if either

- $m > n$ and $a_m > 0$;
- $m = n$ and $a_m \geq b_m$;
- $m < n$ and $b_n < 0$.

Now I claim that $\mathfrak m$ is finitely generated. Indeed, it is generated by $1$: if $p \succ 0$, then $p - 1 \succeq 0$, i.e. $p - 1 \in M$, which means that $p = (p-1) + 1$ is in the ideal generated by $1$.

Finally, I have to exhibit an infinite chain of prime ideals. For each $n \in \mathbb N$, let $I_n$ be the set of polynomials $p$ of degree $\geq n$. If $p, q \succeq 0$, then
$$\deg(p+q) = \max(\deg(p),\deg(q)),\tag{1}$$
because there is no cancellation of leading terms (both are positive). In the case that $\deg p \geq n$, we find $\deg(p+q) \geq n$. This shows that $I_n$ is an ideal. The formula (1) also shows that $I_n$ is prime. This gives an infinite chain of prime ideals
\begin{equation}
I_0 \supseteq I_1 \supseteq I_2 \supseteq \ldots\tag*{$\square$}
\end{equation}

**Remark.** By identifying $k\left[\mathbb Z[x]\right]$ with $k\left[x_0^{\pm 1},x_1^{\pm 1},\ldots\right]$, we can write $R$ as a suitable localisation of the ring
$$k\left[\left\{x_0^{a_0} \cdots x_n^{a_n}\ \big|\ a_n > 0\right\}\right] \subseteq k[x_0^{\pm 1},x_1^{\pm 1}, \ldots].$$
I don't see a neater way to write down more explicitly what this ring is.