Let $(R, \mathfrak m)$ be a local ring such that $\mathfrak m$ is not finitely generated.
Then, must it be true that $\mathfrak m$ is not the radical of a finitely generated ideal ?
I am mostly interested in the case when $R$ is either a Valuation ring https://en.m.wikipedia.org/wiki/Valuation_ring or a ring of prime characteristic $p>0$ which is Perfect (i.e. $R$ is reduced and $R=\{r^p : r \in R \}$ ) .
Note that if $(R, \mathfrak m)$ is a Valuation ring such that $\mathfrak m$ is not finitely generated then $\mathfrak m =\mathfrak m ^2$ .
Please help.