On the ϕ-weak Global Dimensions of Polynomial Rings and ϕ-Prufer Rings

This paper focuses on the study of [Formula: see text]-weak global dimensions in the context of polynomial rings and [Formula: see text]-Prüfer rings. We explore new properties of these dimensions and extend the Hilbert syzygy theorem to [Formula: see text]-weak global dimensions of rings. We also determine the [Formula: see text]-weak global dimension for certain types of [Formula: see text]-Prüfer rings. Key concepts such as [Formula: see text]-flat modules, [Formula: see text]-injective modules, and [Formula: see text]-torsion modules are discussed, along with their hereditary properties in PN-rings. This paper includes several theorems and lemmas that provide insights into the [Formula: see text]-weak global dimensions and their implications in the field of ring theory.