| Popis: |
Let R be a noncommutative prime ring of characteristic not 2 with extended centroid C, the maximal right ring of quotients Q and a nonzero generalized skew derivation $$\delta $$ . Assume that $$f(X_{1},\ldots ,X_{n})$$ is a multilinear polynomial over C that is not central-valued on R and f(R) is the set of all evaluations of the multilinear polynomial $$f\big (X_{1},\ldots ,X_{n}\big )$$ in R. Denote the set $$S:= \big \{\delta (u)u \mid u \in f(R) \big \}$$ . The goal of the paper is to study $$C_R(S)$$ , the centralizer of S in R. To be precise, given a noncentral element $$b\in R$$ it is proved that if $$b\in C_R(S)$$ , i.e., $$\begin{aligned}{}[\delta (u)u, b]=0 \end{aligned}$$ for all $$u \in f(R)$$ , then there exists $$a \in Q$$ with $$[a, b]=0$$ such that $$\delta (x)=ax$$ for all $$x\in R$$ and $$f\big (X_{1},\ldots ,X_{n}\big )^{2}$$ is central-valued on R. As applications to the theorem, we consider the case of $$\delta (u)u \in C$$ for all $$u \in f(R)$$ and also we investigate commuting values of two generalized skew derivations having different associated skew derivations on multilinear polynomials. |
