Reduction of the Finite Generation Problem for Leading Term Ideals under Arbitrary Rational Monomial Orders to the Lexicographic

A commutative ring $R$ is called a Gr\"obner ring if, for every $n\geq 1$, the leading term ideal of every finitely generated ideal of $R[X_1,\ldots,X_n]$ is finitely generated with respect to the lexicographic order $X_1\succ\cdots\succ X_n$. We prove that this property is equivalent to the condition that, for every $n\geq 1$, every rational monomial order $\prec$ on $R[X_1,\ldots,X_n]$, and every finitely generated ideal $I\subseteq R[X_1,\ldots,X_n]$, the leading term ideal $\operatorname{LT}_{\prec}(I)$ is finitely generated. The construction uses a tagged monomial embedding, a compatible group grading, and dehomogenization. As an application, we apply this reduction to valuation rings and determine when the finite generation property holds for every rational monomial order. In particular, for valuation domains, this gives a proof of the rational monomial order version of the Gr\"obner ring conjecture.