AI 中文总结
研究将任意有理单项式序下首项理想的有限生成问题归约到字典序，通过带标签的单项式嵌入等方法证明相关性质等价，还应用此归约于赋值环，给出格罗贝纳环猜想有理单项式序版本的证明。
AI 中文摘要
一个交换环\(R\)被称为格罗贝纳环，如果对于每个\(n\geq1\)，\(R[X_1,\ldots,X_n]\)中每个有限生成理想的首项理想关于字典序\(X_1\succ\cdots\succ X_n\)是有限生成的。我们证明这个性质等同于对于每个\(n\geq1\)，\(R[X_1,\ldots,X_n]\)上的每个有理单项式序\(\prec\)以及每个有限生成理想\(I\subseteq R[X_1,\ldots,X_n]\)，首项理想\(\operatorname{LT}_{\prec}(I)\)是有限生成的条件。构造使用了带标签的单项式嵌入、兼容的群分次和去齐次化。作为应用，我们将此归约应用于赋值环并确定对于每个有理单项式序何时有限生成性质成立。特别地，对于赋值域，这给出了格罗贝纳环猜想的有理单项式序版本的证明。
英文摘要
A commutative ring $R$ is called a Gröbner 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 Gröbner ring conjecture.
