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 Bezout domain of Krull dimension <= 1 if R is a Bezout domain of Krull dimension <= 1. Consequently, defining by induction R(X-1, ... , X-n) := (R(X-1, ... , Xn-1))(X-n), the ring R(X-1, ... , X-n) is a Bezout domain of Krull dimension <= 1 if so is R. The fact that R(X-1, . . . , X-n) is a Bezout 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 Grobner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain R of Krull dimension <= 1, any n E 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 Bezout domain of Krull dimension <= 1 when R is a Bezout domain of Krull dimension <= 1 to any rational monomial order, bolstering the evidence for the Grobner Ring Conjecture in the rational case. We give an example showing that this result is no more true in the irrational case. (c) 2023 Elsevier Inc. All rights reserved.