A negative answer to a question about leading terms ideals of polynomial ideals

We construct an example of a finitely generated ideal $I$ of $R\left[X\right]$, where $R$ is a one-dimensional domain, whose leading terms ideal is not finitely generated. This gives a negative answer to the open question of whether if $R$ is a domain with Krull dimension ≤1, then for any finitely generated ideal $I$ of $R\left[X\right]$, the leading terms ideal of $I$ is also finitely generated. Moreover, as a positive part of our answer, we prove that for any one-dimensional domain $R$ and any $a,b\in R$, the ideal of $R\left[X\right]$ generated by the leading terms of $〈1+aX,b〉$ is finitely generated.