  A code $C = \Phi(\mathcal{C})$ is called $\mathbb{Z}_p
\mathbb{Z}_{p^2}$-linear if it's the Gray image of the $\mathbb{Z}_p
\mathbb{Z}_{p^2}$-additive code $\mathcal{C}$. In this paper, the rank and the
dimension of the kernel of $\mathcal{C}$ are studied. Both of the codes
$\langle \Phi(\mathcal{C}) \rangle$ and $\ker(\Phi(\mathcal{C}))$ are proven
$\mathbb{Z}_p \mathbb{Z}_{p^2}$-additive cyclic codes, and their generator
polynomials are determined. Finally, accurate values of rank and the dimension
of the kernel of some classes of $\mathbb{Z}_p \mathbb{Z}_{p^2}$-additive
cyclic codes are considered.
