The trailing terms ideal over a valuation domain

We prove that if a valuation domain V has Krull dimension ≤1 then for every finitely generated ideal I of V[X] the trailing terms ideal tt(I) of I (that is, the ideal generated by the trailing terms of the nonzero polynomials in I) is also finitely generated. Heinzer and Papick (1989) and then Lombardi, Schuster and Yengui (2012) have shown the analogous result for leading terms ideals. The proof we give is both simple and constructive. The same result is valid for Prüfer domains.