Let $(R, \mathfrak m)$ be a Valuation ring , of finite Krull dimension say $d$, such that $\mathfrak m$ is not principal, hence $\mathfrak m$ is not finitely generated and $\mathfrak m^2=\mathfrak m$ .
Then can we find a principal ideal $J=(r)$ in $R$ such that height $(J)=d$ ? i.e. can we find $r \in R $ such that $\mathfrak m$ is the only prime ideal containing $r$ ?