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Vanishing Derivations on Some Subsets in Prime Rings Involving Generalized Derivations

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Abstract

Let R be a noncommutative prime ring of characteristic different from 2 and 3, C the extended centroid of R, F and G two generalized derivations of R, d a nonzero derivation of R and \(f(x_1,\ldots ,x_n)\) a multilinear polynomial over C. Suppose that I is a nonzero ideal of R and \(f(I)=\{f(x_1,\ldots ,x_n)| x_1,\ldots ,x_n\in I\}\). If \(f(x_1,\ldots ,x_n)\) is not central valued on R and

$$\begin{aligned} d(F^2(u)u-G(u^2))=0 \end{aligned}$$

for all \(u\in f(I)\), then all possible forms of the maps F, G and d are described in the present paper.

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Acknowledgements

The third author expresses his thanks to the University Grants Commission, New Delhi, for its JRF awarded to him under Grant No. F.No. 16-9(June 2019)/2019(NET/CSIR) dated 16.12.2019.

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Correspondence to Basudeb Dhara.

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Communicated by Rosihan M. Ali.

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Dhara, B., De Filippis, V. & Bera, M. Vanishing Derivations on Some Subsets in Prime Rings Involving Generalized Derivations. Bull. Malays. Math. Sci. Soc. 44, 2693–2714 (2021). https://doi.org/10.1007/s40840-021-01080-4

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