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Every unit $u$ in the ring ${𝐙}_n$ of the residual classes mod $n$ induces canonically an automorphism $\pi$ of the algebra ${𝐑}_n(q)=GF(q)[z]/(z^n-1)$. Let $𝒞\subset {𝐑}_n(q)$ be a cyclic code, i.e. an ideal. If the numbers $n$ and $q$ are relatively prime then there exists a well-known characterization of the code $\pi (𝒞)$. We extend this characterization to the general case.

Every unit $u$ in the ring ${𝐙}_n$ of the residual classes mod $n$ induces canonically an automorphism $\pi$ of the algebra ${𝐑}_n(q)=GF(q)[z]/(z^n-1)$. Let $𝒞\subset {𝐑}_n(q)$ be a cyclic code, i.e. an ideal. If the numbers $n$ and $q$ are relatively prime then there exists a well-known characterization of the code $\pi (𝒞)$. We extend this characterization to the general case.