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## Annihilator of Generalized Derivations with Power Values in Rings and Algebras

#### Md Hamidur RAHAMAN [1]

Let $\mathcal{F}, \mathcal{G}$ be two  generalized derivations of prime ring $\mathcal{R}$ with characteristic different from 2 with associated derivations $d_1$ and $d_2$ respectively. We use the symbols  $\mathcal{C}=\mathcal{Z(U)}$ and  $\mathcal{U}$ to denote the  the extended centroid of $R$ and Utumi ring of quotient of $\mathcal{R}$ respectively. Let $0\neq a \in \mathcal{R}$ and $\mathcal{F}$ and $\mathcal{G}$ satisfy $a\{(\mathcal{F}(xy)+\mathcal{G}(yx))^m-[x,y]^n\}=0$ for all $x, y\in \mathcal{J}$, a nonzero ideal, where $m$ and $n$ are natural numbers. Then either $\mathcal{R}$ is commutative or there exists $c$, $b\in \mathcal{U}$ such that $\mathcal{F}$(x) = cx and $\mathcal{G}$(x) = bx for all x ∈ R.
Semiprime rings, Generalized derivations, extended centroid
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Primary Language en Mathematics Articles Orcid: 0000-0003-1822-7863Author: Md Hamidur RAHAMAN (Primary Author)Institution: Aligarh Muslim UniversityCountry: India Publication Date : October 15, 2020
 Bibtex @research article { mathenot631172, journal = {Mathematical Sciences and Applications E-Notes}, issn = {}, eissn = {2147-6268}, address = {}, publisher = {Murat TOSUN}, year = {2020}, volume = {8}, pages = {65 - 70}, doi = {10.36753/mathenot.631172}, title = {Annihilator of Generalized Derivations with Power Values in Rings and Algebras}, key = {cite}, author = {Rahaman, Md Hamidur} } APA Rahaman, M . (2020). Annihilator of Generalized Derivations with Power Values in Rings and Algebras . Mathematical Sciences and Applications E-Notes , 8 (2) , 65-70 . DOI: 10.36753/mathenot.631172 MLA Rahaman, M . "Annihilator of Generalized Derivations with Power Values in Rings and Algebras" . Mathematical Sciences and Applications E-Notes 8 (2020 ): 65-70 Chicago Rahaman, M . "Annihilator of Generalized Derivations with Power Values in Rings and Algebras". Mathematical Sciences and Applications E-Notes 8 (2020 ): 65-70 RIS TY - JOUR T1 - Annihilator of Generalized Derivations with Power Values in Rings and Algebras AU - Md Hamidur Rahaman Y1 - 2020 PY - 2020 N1 - doi: 10.36753/mathenot.631172 DO - 10.36753/mathenot.631172 T2 - Mathematical Sciences and Applications E-Notes JF - Journal JO - JOR SP - 65 EP - 70 VL - 8 IS - 2 SN - -2147-6268 M3 - doi: 10.36753/mathenot.631172 UR - https://doi.org/10.36753/mathenot.631172 Y2 - 2020 ER - EndNote %0 Mathematical Sciences and Applications E-Notes Annihilator of Generalized Derivations with Power Values in Rings and Algebras %A Md Hamidur Rahaman %T Annihilator of Generalized Derivations with Power Values in Rings and Algebras %D 2020 %J Mathematical Sciences and Applications E-Notes %P -2147-6268 %V 8 %N 2 %R doi: 10.36753/mathenot.631172 %U 10.36753/mathenot.631172 ISNAD Rahaman, Md Hamidur . "Annihilator of Generalized Derivations with Power Values in Rings and Algebras". Mathematical Sciences and Applications E-Notes 8 / 2 (October 2020): 65-70 . https://doi.org/10.36753/mathenot.631172 AMA Rahaman M . Annihilator of Generalized Derivations with Power Values in Rings and Algebras. Math. Sci. Appl. E-Notes. 2020; 8(2): 65-70. Vancouver Rahaman M . Annihilator of Generalized Derivations with Power Values in Rings and Algebras. Mathematical Sciences and Applications E-Notes. 2020; 8(2): 65-70. IEEE M. Rahaman , "Annihilator of Generalized Derivations with Power Values in Rings and Algebras", Mathematical Sciences and Applications E-Notes, vol. 8, no. 2, pp. 65-70, Oct. 2020, doi:10.36753/mathenot.631172

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