Some results on \begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document}-additive cyclic codes

\begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document} -Additive cyclic codes of length \begin{document}$ (\alpha,\beta) $\end{document} can be viewed as \begin{document}$ R[x] $\end{document} -submodules of \begin{document}$ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $\end{document} , where \begin{document}$ R = \mathbb{Z}_p+v\mathbb{Z}_p $\end{document} with \begin{document}$ v^2 = v $\end{document} . In this paper, we determine the generator polynomials and the minimal generating sets of this family of codes as \begin{document}$ R[x] $\end{document} -submodules of \begin{document}$ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $\end{document} . We also determine the generator polynomials of the dual codes of \begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document} -additive cyclic codes. Some optimal \begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document} -linear codes and MDSS codes are obtained from \begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document} -additive cyclic codes. Moreover, we also get some quantum codes from \begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document} -additive cyclic codes.