  In this paper, we introduce the concept of $\Sigma$-semicommutative ring, for
$\Sigma$ a finite family of endomorphisms of a ring $R$. We relate this class
of rings with other classes of rings such that Abelian, reduced,
$\Sigma$-rigid, nil-reversible and rings satisfying the $\Sigma$-skew reflexive
nilpotent property. Also, we study some ring-theoretical properties of skew PBW
extensions over $\Sigma$-semicommutative rings. We prove that if a ring $R$ is
$\Sigma$-semicommutative with certain conditions of compatibility on
derivations, then for every skew PBW extension $A$ over $R$, $R$ is Baer if and
only if $R$ is quasi-Baer, and equivalently, $A$ is quasi-Baer if and only if
$A$ is Baer. Finally, we consider some topological conditions for skew PBW
extensions over $\Sigma$-semicommutative rings.
