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

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula> be a prime number and id="M4">\begin{document}$ r, s, t positive integers such that id="M5">\begin{document}$ r\le s\le $\end{document}</tex-math></inline-formula>. A id="M6">\begin{document}$ \mathbb{Z}_{p^r}\mathbb{Z}_{p^s}\mathbb{Z}_{p^t} $\end{document}</tex-math></inline-formula>-additive code is id="M7">\begin{document}$ \mathbb{Z}_{p^t} $\end{document}</tex-math></inline-formula>-submodule of id="M8">\begin{document}$ \mathbb{Z}_{p^r}^{\alpha} \times \mathbb{Z}_{p^s}^{\beta} \mathbb{Z}_{p^t}^{\gamma} $\end{document}</tex-math></inline-formula>, where id="M9">\begin{document}$ \alpha, \beta, \gamma are integers. In this paper, we study id="M10">\begin{document}$ cyclic codes. fact, show these codes can identified as submodules the ring id="M11">\begin{document}$ R = \mathbb{Z}_{p^r}[x]/\big<x^\alpha-1\big> \mathbb{Z}_{p^s}[x]/\big<x^\beta-1\big> \mathbb{Z}_{p^t}[x]/\big<x^\gamma-1\big> Furthermore, determine generator polynomials minimum generating sets kind Moreover, investigate their dual codes.</p>