# Institute of Mathematics

## Generalized derivations acting on multilinear polynomials in prime rings

#### Basudeb Dhara

###### Received July 6, 2016.   First published December 5, 2017.

Abstract:  Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$, let $F$, $G$ and $H$ be three generalized derivations of $R$, $I$ an ideal of $R$ and $f(x_1,\ldots,x_n)$ a multilinear polynomial over $C$ which is not central valued on $R$. If $F(f(r))G(f(r))=H(f(r)^2)$ for all $r=(r_1,\ldots,r_n) \in I^n$, then one of the following conditions holds: \item{(1)} there exist $a\in C$ and $b\in U$ such that $F(x)=ax$, $G(x)=xb$ and $H(x)=xab$ for all $x\in R$; \item{(2)} there exist $a, b\in U$ such that $F(x)=xa$, $G(x)=bx$ and $H(x)=abx$ for all $x\in R$, with $ab\in C$; \item{(3)} there exist $b\in C$ and $a\in U$ such that $F(x)=ax$, $G(x)=bx$ and $H(x)=abx$ for all $x\in R$; \item{(4)} $f(x_1,\ldots,x_n)^2$ is central valued on $R$ and one of the following conditions holds: \itemitem{(a)} there exist $a,b,p,p'\in U$ such that $F(x)=ax$, $G(x)=xb$ and $H(x)=px+xp'$ for all $x\in R$, with $ab=p+p'$; \itemitem{(b)} there exist $a,b,p,p'\in U$ such that $F(x)=xa$, $G(x)=bx$ and $H(x)=px+xp'$ for all $x\in R$, with $p+p'=ab\in C$.
Keywords:  prime ring; derivation; generalized derivation; extended centroid; Utumi quotient ring
Classification MSC:  16W25, 16N60
DOI:  10.21136/CMJ.2017.0352-16

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Affiliations:   Basudeb Dhara, Department of Mathematics, Belda College, Belda, Paschim Medinipur, 721424, West Bengal, India, e-mail: basu_dhara@yahoo.com

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