Identities involving b-generalized skew derivations in prime rings acting as Jordan derivation

Abstract

Let $\mathcal{R}$ symbolize a prime ring with a characteristic other than 2, and let $\rho ({\eta }_1,\dots ,{\eta }_n)$ represent a non-central multilinear polynomial over the extended centroid $\mathcal{C}$ of $\mathcal{R}$. Assuming the presence of three b-generalized skew derivations on $\mathcal{R}$, namely ${\mathcal{D}}_1,{\mathcal{D}}_2$, and ${\mathcal{D}}_3$ that satisfy the equation ${\mathcal{D}}_1(\rho )\rho +\rho {\mathcal{D}}_2(\rho )={\mathcal{D}}_3({\rho }^2)$ for all $\rho =\rho ({\eta }_1,\dots ,{\eta }_n)$, where ${\eta }_1,\dots ,{\eta }_n$ are elements of $\mathcal{R}$. In this research paper, our goal is to elucidate all potential configurations of ${\mathcal{D}}_1,{\mathcal{D}}_2$, and ${\mathcal{D}}_3$.

Keywords:

• b-Generalized derivation
• differential polynomial identity
• multilinear polynomial
• prime ring
• Utumi quotient ring

2020 Mathematics Subject Classification:

• 16N60
• 16W25

Acknowledgments

The author would like to express their sincere thanks to the reviewers and referees for the constructive comments and suggestions which help to improve the quality of the paper.

Disclosure statement

Authors do not have any conflict of interests. All authors are equally contributed.

Additional information

Funding

Second author is supported by DST-INSPIRE (IF190599), Ref. No. DST/INSPIRE/03/2019/000711.