Action of generalized derivations with central values in prime rings

In this paper we are going to show that derivations satisfying some identity carry a certain form. To prove this, we assume \(\mathcal {R}\) is a prime ring with \(char(\mathcal {R})\ne 2\), \(\mathcal {I}\) is a nonzero ideal of \(\mathcal {R}\), \(\mathcal {U}\) is the Utumi quotient ring of \(\mathcal {R}\) with extended centroid \(\mathcal {C}=\mathcal {Z}(\mathcal {U})\) and \(f(x_1,\ldots ,x_n)\) is any noncentral valued multilinear polynomial over \(\mathcal {C}\). Suppose that \(\mathcal {F}\) and \(\mathcal {G}\) are two generalized derivations and d is any non-zero derivation of \(\mathcal {R}\). If

$$\begin{aligned}\mathcal {F}^2(f(\zeta ))d(f(\zeta ))-\mathcal {G}(f(\zeta )^2) \in \mathcal {C}\end{aligned}$$

for all \(\zeta =(\zeta _1,\ldots ,\zeta _n)\in \mathcal {I}^n\), then \(\mathcal {F}(x)=ax\) or \(\mathcal {F}(x)=xa\) for any \(x\in \mathcal {R}\), for some \(a\in \mathcal {U}\) along with \(a^2 =0\) and following one conclusion holds:

1. (1)

   \(\mathcal {G}=0\);

2. (2)

   there exists \(\lambda \in \mathcal {C}\) and a derivation h of \(\mathcal {R}\) such that \(\mathcal {G}(x)= \lambda x+h(x)\) for all \(x\in \mathcal {R}\) with \(f(\zeta _1,\ldots ,\zeta _n)^2\) is central valued on \(\mathcal {R}\);

3. (3)

   \(\mathcal {R}\) satisfies \(s_4\).