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

Abstract

This paper explores the dimension theory of non-Noetherian graded rings by introducing the class of Hilbert--Serre rings. We generalize Krull's dimension theorem and Smoke's dimension theorem by establishing the fundamental inequalities $\dim_{\mathrm{gr}}(R) \leq \dim(R) \leq \operatorname{GKdim}_k(R) \leq d(R)$ for any Hilbert--Serre ring $R$, where $d(R)$ is the pole order of its Poincaré series at $t=1$. Furthermore, we apply these results to initial algebras, proving that all these dimensions, including the transcendence degree, coincide for monomial algebras. Finally, we provide explicit examples demonstrating that these inequalities can be strict in general, even for integral domains.