Derivations vanishing on commutator identity involving generalized derivation on multilinear polynomials in prime rings

Abstract

Let R be a prime ring of characteristic different from 2 with its Utumi quotient ring U and extended centroid C, $f(x_1,\dots ,x_n)$ be multilinear polynomial over C, which is not central valued on R. If d is a non-zero derivation of R and F is a non-zero generalized derivation of R such that $$d\left(\right[F\left(f\right(r_1,\dots ,r_n\left)\right)f(r_1,\dots ,r_n),F\left(f\right(s_1,\dots ,s_n\left)\right)f(s_1,\dots ,s_n)\left]\right)=0$$

for all $r_1,\dots ,r_n,s_1,\dots ,s_n\in R$, then there exists $c\in U$ such that F(x) = cx, for all $x\in R$ and one of the following holds:

• (1) $f{(x_1,\dots ,x_n)}^2$ is central valued on R;

• (2) R satisfies s₄, the standard identity of degree 4.

Keywords:

• Generalized derivation
• prime ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16N20
• 16W25