Rather large subsets and vanishing generalized derivations on multilinear polynomials

ABSTRACT

Let R be a prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid and let $f(x_1,\dots ,x_n)$ be a multilinear polynomial over C, not central valued on R. Suppose that F and G are non-zero generalized derivations of R and 0≠u₀ is an element of R such that

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{u_{0}F\left(G\left(f\right(r_1,\dots ,r_n\left)\right)f(r_1,\dots ,r_n)\right)=0}\end{array}}$

for all $r_1,\dots ,r_n\in R$. Then one of the following holds:

1. there exists a,c∈U, the right Utumi quotient ring of R, such that F(x) = ax and G(x) = cx, for all x∈R, with u₀ac = 0;

2. there exists a∈U, the right Utumi quotient ring of R, such that F(x) = ax, for all x∈R, with u₀a = 0;

3. there exists a,b,c∈U, the right Utumi quotient ring of R, such that F(x) = ax+xb and G(x) = cx, for all x∈R, with $u_{0}c=u_{0}ac=0$;

4. $f{(x_1,\dots ,x_n)}^2$ is central valued on R and there exists a,b,c∈U, such that F(x) = ax+xb, G(x) = cx, for all x∈R, with u₀(ac+cb) = 0;

5. there exists a,c∈U and d:R→R a derivation of R such that F(x) = ax+d(x) and G(x) = cx, for all x∈R, with $u_{0}c=u_0(ac+d\left(c\right))=0$. Moreover, in this case, d is not an inner derivation of R.

KEYWORDS:

• Generalized skew derivation
• Lie ideal
• prime ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W25
• 16N60