$\mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}{[u^2]}$-additive skew cyclic codes of length $2p^s $

In this paper, we first study the skew cyclic codes of length \begin{document}$ p^s $\end{document} over \begin{document}$ R_3 = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}, $\end{document} where \begin{document}$ p $\end{document} is a prime number and \begin{document}$ u^3 = 0. $\end{document} Then we characterize the algebraic structure of \begin{document}$ \mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}[u^2] $\end{document} -additive skew cyclic codes of length \begin{document}$ 2p^s. $\end{document} We will show that there are sixteen different types of these codes and classify them in terms of their generators.