On Skew Polynomials over p.q.-Baer and p.p.-Modules

Let $M_R$ be a module and $\sigma$ an endomorphism of $R$. Let $m\in M$ and $a\in R$, we say that $M_R$ satisfies the condition ${\mathcal{C}}_1$ (respectively, ${\mathcal{C}}_2$), if $ma=0$ implies $m\sigma (a)=0$ (respectively, $m\sigma (a)=0$ implies $ma=0$). We show that if $M_R$ is p.q.-Baer then so is $M[x;\sigma {]}_{R[x;\sigma ]}$ whenever $M_R$ satisfies the condition ${\mathcal{C}}_2$, and the converse holds when $M_R$ satisfies the condition ${\mathcal{C}}_1$. Also, if $M_R$ satisfies ${\mathcal{C}}_2$ 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 $M[x,x^{-1};\sigma {]}_{R[x,x^{-1};\sigma ]}$ ($\sigma \in Aut(R)$) is a p.p.-module. Many generalizations are obtained, and more results are found when $M_R$ is a semicommutative module.