On the $\varphi$-weak global dimensions of polynomial rings and $\varphi$-Prüfer rings

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