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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]

There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics. [ABSTRACT FROM AUTHOR]