Dhara, B., Kar, S., Pradhan, K. (2016). Co-centralizing generalized derivations acting on multilinear polynomials in prime rings. Bulletin of the Iranian Mathematical Society, 42(6), 1331-1342.

B. Dhara; S. Kar; K. G. Pradhan. "Co-centralizing generalized derivations acting on multilinear polynomials in prime rings". Bulletin of the Iranian Mathematical Society, 42, 6, 2016, 1331-1342.

Dhara, B., Kar, S., Pradhan, K. (2016). 'Co-centralizing generalized derivations acting on multilinear polynomials in prime rings', Bulletin of the Iranian Mathematical Society, 42(6), pp. 1331-1342.

Dhara, B., Kar, S., Pradhan, K. Co-centralizing generalized derivations acting on multilinear polynomials in prime rings. Bulletin of the Iranian Mathematical Society, 2016; 42(6): 1331-1342.

Co-centralizing generalized derivations acting on multilinear polynomials in prime rings

Receive Date: 04 April 2015,
Revise Date: 25 August 2015,
Accept Date: 26 August 2015

Abstract

Let $R$ be a noncommutative prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C$ $(=Z(U))$ the extended centroid of $R$. Let $0\neq a\in R$ and $f(x_1,\ldots,x_n)$ a multilinear polynomial over $C$ which is noncentral valued on $R$. Suppose that $G$ and $H$ are two nonzero generalized derivations of $R$ such that $a(H(f(x))f(x)-f(x)G(f(x)))\in C$ for all $x=(x_1,\ldots,x_n)\in R^n$. one of the following holds: $f(x_1,\ldots,x_n)^2$ is central valued on $R$ and there exist $b,p,q\in U$ such that $H(x)=px+xb$ for all $x\in R$, $G(x)=bx+xq$ for all $x\in R$ with $a(p-q)\in C$; there exist $p,q\in U$ such that $H(x)=px+xq$ for all $x\in R$, $G(x)=qx$ for all $x\in R$ with $ap=0$; $f(x_1,\ldots,x_n)^2$ is central valued on $R$ and there exist $q\in U$, $\lambda\in C$ and an outer derivation $g$ of $U$ such that $H(x)=xq+\lambda x-g(x)$ for all $x\in R$, $G(x)=qx+g(x)$ for all $x\in R$, with $a\in C$; $R$ satisfies $s_4$ and there exist $b,p\in U$ such that $H(x)=px+xb$ for all $x\in R$, $G(x)=bx+xp$ for all $x\in R$.

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