  It is well-known that for any commutative unitary ring $\mathbf{R}$, the
Serre conjecture ring $\mathbf{R}\langle X \rangle$, i.e., the localization of
the univariate polynomial ring $\mathbf{R}[X]$ at monic polynomials, is a
B\'ezout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$.
Consequently, defining by induction $\mathbf{R}\langle X_1,\ldots,X_n
\rangle:=(\mathbf{R}\langle X_1,\ldots,X_{n-1}\rangle)\langle X_n\rangle$, the
ring $\mathbf{R}\langle X_1,\ldots,X_n \rangle$ is a B\'ezout domain of Krull
dimension $\leq 1$ if so is $\mathbf{R}$. The fact that $\mathbf{R}\langle
X_1,\ldots,X_n \rangle$ is a B\'ezout domain when $\mathbf{R}$ is a valuation
domain of Krull dimension $\leq 1$ was the cornerstone of Brewer and Costa's
theorem stating that if $\mathbf{R}$ is a one-dimensional arithmetical ring
then finitely generated projective $\mathbf{R}[X_1,\dots,X_n]$-modules are
extended. It is also the key of the proof of the Gr\"obner Ring Conjecture in
the lexicographic order case, namely the fact that for any valuation domain
$\mathbf{R}$ of Krull dimension $\leq 1$, any $n \in \mathbb{N}_{>0}$, and any
finitely generated ideal $I$ of $\mathbf{R}[X_1, \dots, X_n]$, the ideal
$\operatorname{LT}(I)$ generated by the leading terms of the elements of $I$
with respect to the lexicographic monomial order is finitely generated. Since
the ring $\mathbf{R}\langle X_1,\ldots,X_n\rangle$ can also be defined directly
as the localization of the multivariate polynomial ring
$\mathbf{R}[X_1,\dots,X_n]$ at polynomials whose leading coefficients according
to the lexicographic monomial order with $X_1<X_2<\cdots<X_n$ is $1$, we
propose to generalize the fact that $\mathbf{R}\langle X_1,\ldots,X_n\rangle$
is a B\'ezout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$ to any
rational monomial order, bolstering the evidence for the Gr\"obner Ring
Conjecture in the rational case.
