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

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