Generalized derivations with multilinear polynomials in prime rings

Abstract

Let R be a prime ring of characteristic different from two with Utumi quotient ring U and extended centroid C, $f(x_1,\dots ,x_n)$ be a multilinear polynomial over C, which is not central valued on R. Suppose that F, G and H are three generalized derivations on R. If $F(f(r))G(f(r))-f(r)H(f(r))=0$ for all $r=(r_1,\dots ,r_n)\in R^n$, then one of the following holds:

(i) G = 0 and H = 0,

(ii) F = 0 and H = 0,

(iii) there exist $a\in C,b,c\in U$ such that F(x)=ax, $G(x)=bx+xc$ and $H(x)=a(bx+xc)$ for all $x\in R$,

(iv) there exist $a,b\in U$ such that F(x)=xa, G(x)=bx and H(x)=abx for all $x\in R$,

(v) there exist $a\in C,b\in U$ such that F(x)=ax, G(x)=xb and H(x)=xab for all $x\in R$,

(vi) $f{(r_1,\dots ,r_n)}^2$ is central valued on R and one of the following holds:

(a) there exist $a,b\in U$ such that F(x)=ax, G(x)=xb and H(x)=xab for all $x\in R$,

(b) there exist $c\in C,a,b\in U$ such that $F(x)=ax+xb$, G(x)=cx and $H(x)=c(bx+xa)$ for all $x\in R$.

Keywords:

• Derivation
• extended centroid
• generalized derivation
• prime ring
• utumi quotient ring

2010 Mathematics Subject Classification:

• 16W25
• 16N60

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 article.