Annihilating co-commutators with generalized skew derivations on multilinear polynomials

ABSTRACT

Let ℛ be a prime ring of characteristic different from 2, 𝒬_r be its right Martindale quotient ring, 𝒬 be its two-sided Martindale quotient ring and 𝒞 be its extended centroid. Suppose that ℱ, 𝒢 are additive mappings from ℛ into itself and that $f(x_1,\dots ,x_n)$ is a non-central multilinear polynomial over 𝒞 with n non-commuting variables. We prove the following results:

(a) If ℱ and 𝒢 are generalized derivations of ℛ such that

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{f\left(\overline{r}\right)\left(\mathcal{ℱ}\right(f\left(\overline{r}\right)\left)f\right(\overline{r})-f(\overline{r}\left)\mathcal{𝒢}\right(f\left(\overline{r}\right)\left)\right)=0}\end{array}}$

for all $\overline{r}\in {\mathcal{ℛ}}^n$, then one of the following holds:

(a) there exists q∈𝒬 such that ℱ(x) = xq and 𝒢(x) = qx for all x∈ℛ.

(b) there exist c,q∈𝒬 such that ℱ(x) = qx+xc, 𝒢(x) = cx+xq for all x∈ℛ, and $f{(x_1,\dots ,x_n)}^2$ is central-valued on ℛ.

(b) If ℱ is a generalized skew derivation of ℛ such that

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{f\left(\overline{r}\right)\left[\mathcal{ℱ}\right(f\left(\overline{r}\right)),f(\overline{r}\left)\right]=0}\end{array}}$

for all $\overline{r}\in {\mathcal{ℛ}}^n$, then one of the following holds:

(a) there exists λ∈𝒞 such that ℱ(x) = λx for all x∈ℛ;

(b) there exist q∈𝒬_r and λ∈𝒞 such that ℱ(x) = (q+λ)x+xq for all x∈ℛ, and $f{(x_1,\dots ,x_n)}^2$ is central-valued on ℛ.

KEYWORDS:

• Generalized skew derivation
• multilinear polynomial
• prime ring

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W25
• 16N60

Acknowledgment

This research was done when the first and second authors visited the School of Mathematics and Statistics at Beijing Institute of Technology in the spring of 2013. They take this opportunity to express their sincere thanks to the School of Mathematics and the Office of International Affairs at Beijing Institute of Technology for the hospitality extended to them during their visit. All authors are deeply grateful to a special Training Program of International Exchange and Cooperation of the Beijing Institute of Technology.