On Krull's Dimension Theorem for Certain Graded Rings and Its Applications

This paper explores the dimension theory of non-Noetherian graded rings by introducing the class of Hilbert–Serre rings. Motivated by Krull’s dimension theorem and Smoke’s dimension theorem, we establish the fundamental inequalities $\mathrm{}(R)\le \dim (R)\le {\mathrm{GKdim}}_k(R)\le d(R)$ for any Hilbert–Serre ring $R$, where $d(R)$ is the pole order of its Poincaré series at $t=1$. We then prove that all these dimensions, including the transcendence degree in the domain case, coincide for monomial algebras. As an application, let $S$ be a finitely generated homogeneous subalgebra of a polynomial ring over a field $k$, and let ${\mathrm{i}\mathrm{n}}_{<}(S)$ be its initial algebra with respect to a monomial order $<$. Even though ${\mathrm{i}\mathrm{n}}_{<}(S)$ need not be Noetherian, we prove that taking initial algebras preserves all the dimensions considered in this paper; in particular, $\dim (S)=\dim ({\mathrm{i}\mathrm{n}}_{<}(S))$. Finally, we provide explicit examples demonstrating that these inequalities can be strict in general, even for Hilbert–Serre domains.