An identity on generalized derivations involving multilinear polynomials in prime rings

Abstract

Let R be a prime ring of characteristic different from 2 with its Utumi ring of quotients U, extended centroid C, $f(x_1,\ldots ,x_n)$ a multilinear polynomial over C, which is not central-valued on R and d a nonzero derivation of R. By f(R), we mean the set of all evaluations of the polynomial $f(x_1,\ldots ,x_n)$ in R. In the present paper, we study $b[d(u),u]+p[d(u),u]q+[d(u),u]c=0$ for all $u\in f(R)$, which includes left-sided, right-sided as well as two-sided annihilating conditions of the set $\{[d(u),u] : u\in f(R)\}$. We also examine some consequences of this result related to generalized derivations and we prove that if F is a generalized derivation of R and d is a nonzero derivation of R such that

$$\begin{aligned} F^2([d(u), u])=0 \end{aligned}$$

for all $u\in f(R)$, then there exists $a\in U$ with $a^2=0$ such that $F(x)=xa$ for all $x\in R$ or $F(x)=ax$ for all $x\in R$.