Generalized derivations act as a Jordan homomorphism on multilinear polynomials

Abstract

Let R be prime ring of characteristic different from 2 and F, G, and H are generalized derivations on R. Let U be Utumi quotient ring of R with extended centroid C and $f(x_1,\dots ,x_n)$ be a multilinear polynomial over C which is not central valued on R. If $$F\left(f\right(r\left)\right)G\left(f\right(r\left)\right)-f\left(r\right)H\left(f\right(r\left)\right)=H\left(f{\left(r\right)}^2\right)$$ for all $r=(r_1,\dots ,r_n)\in R^n$ and give the complete possible forms of the mappings F, G, and H.

Keywords:

• Extended centroid
• generalized derivations
• Jordan homomorphism
• multilinear polynomials
• prime rings
• Utumi quotient ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16N60
• 16W25

Additional information

Funding

First author is supported by the PNPD-CAPES fellowship under the supervision of Prof. Daniel Levcovitz. Second author is supported by the research grant DST-SERB EMR/2016/001550.