Abstract
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 as 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.
