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

AbstractWe construct an example of a finitely generated ideal I of R[X], 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[X], 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∈R, the ideal of R[X] generated by the leading terms of 〈1+aX,b〉 is finitely generated