  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.
