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It is well-known that for any integral domain R , the Serre conjecture ring R 〈 X 〉 , i.e., the localization of the univariate polynomial ring R [ X ] at monic polynomials, is a Bézout domain of Krull dimension ≤1 if R is a Bézout domain of Krull dimension ≤1. Consequently, defining by induction R 〈 X 1 , ... , X n 〉 : = (R 〈 X 1 , ... , X n − 1 〉) 〈 X n 〉 , the ring R 〈 X 1 , ... , X n 〉 is a Bézout domain of Krull dimension ≤1 if so is R. The fact that R 〈 X 1 , ... , X n 〉 is a Bézout domain when R is a valuation domain of Krull dimension ≤1 was the cornerstone of Brewer and Costa's theorem stating that if R is a one-dimensional arithmetical ring then finitely generated projective R [ X 1 , ... , X n ] -modules are extended. It is also the key of the proof of the Gröbner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain R of Krull dimension ≤1, any n ∈ N > 0 , and any finitely generated ideal I of R [ X 1 , ... , X n ] , the ideal LT (I) generated by the leading terms of the elements of I with respect to the lexicographic monomial order is finitely generated. Since the ring R 〈 X 1 , ... , X n 〉 can also be defined directly as the localization of the multivariate polynomial ring R [ X 1 , ... , X n ] at polynomials whose leading coefficients according to the lexicographic monomial order with X 1 < X 2 < ⋯ < X n is 1, we propose to generalize the fact that R 〈 X 1 , ... , X n 〉 is a Bézout domain of Krull dimension ≤1 when R is a Bézout domain of Krull dimension ≤1 to any rational monomial order, bolstering the evidence for the Gröbner Ring Conjecture in the rational case. We give an example showing that this result is no more true in the irrational case. [ABSTRACT FROM AUTHOR]

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]