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## Pair of generalized derivations acting on multilinear polynomials in prime rings

#### Basudeb DHARA [1] , Sukhendu KAR [2] , Priyadwip DAS [3]

Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$ and $f(r_1,\ldots,r_n)$ be a multilinear polynomial over $C$, which is not central valued on $R$. Suppose that $F$ and $G$ are two nonzero generalized derivations of $R$ such that $G\neq Id$ (identity map) and $$F(f(r)^2)=F(f(r))G(f(r))+G(f(r))F(f(r))$$

for all $r=(r_1,\ldots,r_n)\in R^n$. Then one of the following holds:

(1) there exist $\lambda \in C$ and $\mu \in C$ such that $F(x)=\lambda x$ and $G(x)=\mu x$ for
all $x\in R$ with $2\mu=1$;
(2) there exist $\lambda \in C$ and $p,q\in U$ such that $F(x)=\lambda x$ and $G(x)=px+xq$ for all $x\in R$ with $p+q\in C$,
$2(p+q)=1$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;
(3) there exist $\lambda \in C$ and $a\in U$ such that $F(x)=[a,x]$ and $G(x)=\lambda x$ for all $x\in R$ with $f(x_1,\ldots,x_n)^2$ is central valued on $R$;
(4) there exist $\lambda \in C$ and $a,b\in U$ such that $F(x)=ax+xb$ and $G(x)=\lambda x$ for all $x\in R$ with $a+b\in C$, $2\lambda =1$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;
(5) there exist $a, p\in U$ such that $F(x)=xa$ and $G(x)=px$ for all $x\in R$, with $(p-1)a=-ap\in C$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;
(6) there exist $a, q\in U$ such that $F(x)=ax$ and $G(x)=xq$ for all $x\in R$ with $a(q-1)=-qa\in C$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$.

prime ring, derivation, generalized derivation, extended centroid
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Birincil Dil en Matematik Matematik Orcid: 0000-0002-8345-1362Yazar: Basudeb DHARA Kurum: Belda CollegeÜlke: India Orcid: 0000-0002-3955-9464Yazar: Sukhendu KAR Kurum: Jadavpur UniversityÜlke: India Orcid: 0000-0001-9898-6485Yazar: Priyadwip DAS Kurum: Jadavpur UniversityÜlke: India Yayımlanma Tarihi : 2 Nisan 2020
 Bibtex @araştırma makalesi { hujms588747, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe Üniversitesi}, year = {2020}, volume = {49}, pages = {740 - 753}, doi = {10.15672/hujms.588747}, title = {Pair of generalized derivations acting on multilinear polynomials in prime rings}, key = {cite}, author = {DHARA, Basudeb and KAR, Sukhendu and DAS, Priyadwip} } APA DHARA, B , KAR, S , DAS, P . (2020). Pair of generalized derivations acting on multilinear polynomials in prime rings. Hacettepe Journal of Mathematics and Statistics , 49 (2) , 740-753 . DOI: 10.15672/hujms.588747 MLA DHARA, B , KAR, S , DAS, P . "Pair of generalized derivations acting on multilinear polynomials in prime rings". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 740-753 Chicago DHARA, B , KAR, S , DAS, P . "Pair of generalized derivations acting on multilinear polynomials in prime rings". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 740-753 RIS TY - JOUR T1 - Pair of generalized derivations acting on multilinear polynomials in prime rings AU - Basudeb DHARA , Sukhendu KAR , Priyadwip DAS Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.588747 DO - 10.15672/hujms.588747 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 740 EP - 753 VL - 49 IS - 2 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.588747 UR - https://doi.org/10.15672/hujms.588747 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Pair of generalized derivations acting on multilinear polynomials in prime rings %A Basudeb DHARA , Sukhendu KAR , Priyadwip DAS %T Pair of generalized derivations acting on multilinear polynomials in prime rings %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 2 %R doi: 10.15672/hujms.588747 %U 10.15672/hujms.588747 ISNAD DHARA, Basudeb , KAR, Sukhendu , DAS, Priyadwip . "Pair of generalized derivations acting on multilinear polynomials in prime rings". Hacettepe Journal of Mathematics and Statistics 49 / 2 (Nisan 2020): 740-753 . https://doi.org/10.15672/hujms.588747 AMA DHARA B , KAR S , DAS P . Pair of generalized derivations acting on multilinear polynomials in prime rings. Hacettepe Journal of Mathematics and Statistics. 2020; 49(2): 740-753. Vancouver DHARA B , KAR S , DAS P . Pair of generalized derivations acting on multilinear polynomials in prime rings. Hacettepe Journal of Mathematics and Statistics. 2020; 49(2): 753-740.

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