For a finite commutative ring R and a positive integer κ ≥ 2, we construct an iteration digraph G(R, κ) whose vertex set is R and for which there is a directed edge from α ∈ R to b ∈ R if b = α~κ. In this paper, we investigate the iteration digraphs G(F_pr C_n, κ) of F_pr C_n, the group ring of a cyclic group C_n over a finite field F_pr. We study the cycle structure of G(F_pr C_n, κ), and explore the symmetric digraphs. Finally, we obtain necessary and sufficient conditions on F_pr C_n and κ such that G(F_pr C_n, κ) is semiregular.
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