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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 t t (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. [ABSTRACT FROM AUTHOR]

We propose a counterexample to the Gröbner ring conjecture in the irrational case. More precisely, we construct a valuation domain V of Krull dimension 1 and a finitely generated ideal J of V [ X , Y ] equipped with some irrational monomial order <, such that the ideal LT < (J) generated by the leading terms of the elements of J is not finitely generated. [ABSTRACT FROM AUTHOR]