. Also, if $M_R$ satisfies $\mathcal{C}_0$ and $\sigma$-skew Armendariz, then $M_R$ is a p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module if and only if $M[x,x^{-0};\sigma]_{R[x,x^{-0};\sigma]}$ ($\sigma\in Aut(R)$) is a p.p.-module