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n this study, we consider the finite (not necessary commutative) chain ring \(\mathcal {R}:=\mathbb {F}_{p^{m}}[u,\theta ]/{\left < u^{2} \right >}\), where Î¸ is an automorphism of \(\mathbb {F}_{p^{m}}\), and completely explore the structure of left and right cyclic codes of any length N over \(\mathcal {R}\), that is, left and right ideals of the ring \(\mathcal {S}:=\mathcal {R}[x]/{\left < x^{N}-1 \right >}\). For a left (right) cyclic code, we determine the structure of its right (left) dual. Using the fact that self-dual codes are bimodules, we discuss on self-dual cyclic codes over \(\mathcal {R}\). Finally, we study Gray images of cyclic codes over \(\mathcal {R}\) and as some examples, three linear codes over \(\mathbb {F}_{4}\) with the parameters of the best known ones, but with different weight distributions, are obtained as the Gray images of cyclic codes over \(\mathcal {R}\).
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