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arXiv · 2201.08339

$\Sigma$-semicommutative rings and their skew PBW extensions

Héctor Suárez·Armando Reyes
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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 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.

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Héctor Suárez, Armando Reyes. 2022-01-20. $\Sigma$-semicommutative rings and their skew PBW extensions. https://arxiv.org/abs/2201.08339

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