A note on generalized derivations with multilinear polynomials in prime rings

Abstract

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R with the extended centroid C and $f(x_1,\dots ,x_n)$ be a non-central multilinear polynomial over C. Suppose that F and G are two generalized derivations on R such that $F^2(u)u-G(u^2)\in C$ for all $u=f(r_1,\dots ,r_n),$ then we give the complete description of the mappings F and G.

Keywords:

• Derivation
• extended centroid
• generalized derivation
• Prime ring
• Utumi quotient ring

Mathematics Subject Classification 2010:

• 16W25
• 16N60

Acknowledgment

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.

Additional information

Funding

The second author is supported by the Department of Science & Technology (DST), Government of India, Early Career Research Award ECR/2016/000649