In this article, we study twisted derivations of cyclic group rings. Let R be a commutative ring with unity, G be a finite cyclic group, and ( \(\sigma , \tau \) ) be a pair of R-algebra endomorphisms of the group algebra RG, which are R-linear extensions of the group endomorphisms of G. In this article, we give two characterizations concerning \((\sigma , \tau )\) -derivations of the group ring RG. First, we develop a necessary and sufficient condition for a \((\sigma , \tau )\) -derivation of RG to be inner. Second, we provide a necessary and sufficient condition for an R-linear map \(D: RG \rightarrow RG\) with \(D(1) = 0\) to be a \((\sigma , \tau )\) -derivation. We also illustrate our theorems with the help of examples. As a consequence of these two characterizations, we answer the well-known twisted derivation problem for RG: Under what conditions are all \((\sigma , \tau )\) -derivations of RG inner? Or is the space of outer \((\sigma , \tau )\) -derivations trivial? More precisely, we give a sufficient condition under which all \((\sigma , \tau )\) -derivations of RG are inner and a sufficient condition under which RG has non-trivial outer \((\sigma , \tau )\) -derivations. Our result helps in generating several examples of non-trivial outer \((\sigma , \tau )\) -derivations.
