ABSTRACT

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R and $f(x_1,\dots ,x_n)$ a multilinear polynomial over C. Suppose that $f(x_1,\dots ,x_n)$ is not central valued on R, and R does not embed in $M_2\left(K\right)$, the algebra of $2\times 2$ matrices over a field K. We describe all possible forms of F and G in the case $\left[F\right(G\left(f\right(r_1,\dots ,r_n\left)\right)),f(r_1,\dots ,r_n\left)\right]=0$ for any $r_1,\dots ,r_n\in R$.

中文翻译：

---

素环多重线性多项式广义导数的对易积

摘要

令R是特征不同于 2 的素数环，U是R的 Utumi 商环，C是R的扩展质心，F和G的非零广义推导R和$F(X_1,\dots ,X_n)$C上的多项式多项式。假设$F(X_1,\dots ,X_n)$在R上不是中心值，并且R不嵌入${米}_2\left(ķ\right)$, 的代数$2\times 2$域K上的矩阵。我们描述了案例中所有可能的F和G形式$\left[F\right(G\left(F\right(r_1,\dots ,r_n\left)\right)),F(r_1,\dots ,r_n\left)\right]=0$对于任何$r_1,\dots ,r_n\in R$.