On the leading terms ideals of polynomial ideals over a valuation ring

We construct an example of a finitely generated ideal I of V[X], where V is a one-dimensional valuation ring, whose leading terms ideal is not finitely generated. This gives a negative answer to the open question of whether if V is a valuation ring with Krull dimension ⩽1, then for any finitely generated ideal I of V[X], the leading terms ideal of I is also finitely generated. The valuation rings satisfying this latter property will be called 1-Gröbner and are studied in this paper.