A note on the Jacobson radical of Ore extensions

Abstract

Let R be a ring with a monomorphism α and an α-derivation δ. In this article, we give a simple and different proof about the semiprimitivity of Ore extensions which states that the skew polynomial ring $R[x;\alpha ,\delta ]$ is semiprimitive reduced if and only if R is α-rigid. This unifies and extends a number of known results on the Jacobson radical in the special cases. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of rings whose the set of all nilpotent elements has special conditions. Important examples are provided to illustrate the applications of the results.

Keywords:

• Jacobson radical
• NI ring
• Ore extension
• semiprimitive ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16N20
• 16N40
• 16S36

Authors’ contributions

All authors contribute to finding and proving the results. The second author wrote the manuscript and the other authors proof read the manuscript.

Data availability statement

The results have been obtained with the help of the existing articles mentioned in the text.

Disclosure statement

The authors declare that there are no conflict of interests in the manuscript.

Ethical approval

Not applicable.

Additional information

Funding

No specific funding has been received.