$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $-additive cyclic codes

Let \begin{document}$ p $\end{document} be a prime number and \begin{document}$ r, s, t $\end{document} be positive integers such that \begin{document}$ r\le s\le t $\end{document}. A \begin{document}$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $\end{document}-additive code is a \begin{document}$ \mathbb{Z}_{p^t} $\end{document}-submodule of \begin{document}$ \mathbb{Z}_{p^r}^{\alpha} \times \mathbb{Z}_{p^s}^{\beta} \times \mathbb{Z}_{p^t}^{\gamma} $\end{document}, where \begin{document}$ \alpha, \beta, \gamma $\end{document} are positive integers. In this paper, we study \begin{document}$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $\end{document}-additive cyclic codes. In fact, we show that these codes can be identified as submodules of the ring \begin{document}$ R = \mathbb{Z}_{p^r}[x]/\big<x^\alpha-1\big> \times \mathbb{Z}_{p^s}[x]/\big<x^\beta-1\big> \times \mathbb{Z}_{p^t}[x]/\big<x^\gamma-1\big> $\end{document}. Furthermore, we determine the generator polynomials and minimum generating sets of this kind of codes. Moreover, we investigate their dual codes.