2026/07/28 by Xiaopeng Zheng
#math.AC
A commutative ring R is called a Gröbner ring if, for every n≥ 1, the leading term ideal of every finitely generated ideal of R[X1,…,Xn] is finitely generated with respect to the lexicographic order X1\succ⋯\succ Xn. We prove that this property is equivalent to the condition that, for every n≥ 1, every rational monomial order \prec on R[X1,…,Xn], and every finitely generated ideal I⊆ R[X1,…,Xn], the leading term ideal 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.