Let $\C$ be a cyclic group of order $q$ and $n$ is a divisor of $Q$, while $r$ is a divisor of $Q/n$; then under some restrictions, an element $\alpha\in\mathfrak{C}_Q$ is called $(r,n)$-free. Similarly, if $f,g\in\mathbb{F}_q[x]$ are divisors of $x^m-1$, then under special conditions, an element $\alpha\in\mathbb{F}_{q^m}$ is called $(f,g)$-free element. We generalize the notions $(r,n)$-free and $(f,g)$-free elements of a finite cyclic module and establish characteristic functions for these elements.