Characteristic functions for \MakeLowercase{(r, n)}-free and \MakeLowercase{(f, g)}-free elements

For a prime power $q$, $\text{F}$ denotes the finite field of order $q$, and for $m\ge 2$, $\text{Fm}$ denotes the extension field of degree $m$. We establish a characteristic function for the set of $(r,n)$-free elements of finite cyclic $R$-module for the Euclidean domain $R$. Furthermore, we explore $(f,g)$-freeness through polynomial values and finally give an expression for the characteristic function for the set of $(f,g)$-free elements.