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\(\Sigma\)-Semicommutative rings and their skew PBW extensions

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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 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.

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Acknowledgements

The first author was supported by Vicerrectoría de Investigación y Extensión, Code SGI 3334, Universidad Pedagógica y Tecnológica de Colombia, Tunja, Colombia. The second author was supported by the research fund of Faculty of Science, Code HERMES 52464, Universidad Nacional de Colombia - Sede Bogotá, Colombia.

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Authors and Affiliations

  1. Escuela de Matemáticas y Estadística, Grupo de Álgebra y Análisis, Universidad Pedagógica y Tecnológica de Colombia - Sede Tunja, Tunja, Colombia

    Héctor Suárez

  2. Campus Universitario, Tunja, Colombia

    Héctor Suárez & Armando Reyes

  3. Departamento de Matemáticas, Universidad Nacional de Colombia - Sede Bogotá, Bogotá, Colombia

    Armando Reyes

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  1. Héctor Suárez
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Correspondence to Héctor Suárez.

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Communicated by Iván Ezequiel Angiono.

Dedicated to the memory of Professor V. A. Artamonov.

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Suárez, H., Reyes, A. \(\Sigma\)-Semicommutative rings and their skew PBW extensions. São Paulo J. Math. Sci. 17, 531–554 (2023). https://doi.org/10.1007/s40863-023-00356-w

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