Generalized Derivations Acting on Multilinear Polynomials in Prime Rings and Banach Algebras

Basudeb Dhara , Nurcan Argaç

Communications in Mathematics and Statistics ›› 2016, Vol. 4 ›› Issue (1) : 39 -54.

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Communications in Mathematics and Statistics ›› 2016, Vol. 4 ›› Issue (1) : 39 -54. DOI: 10.1007/s40304-015-0073-y
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Generalized Derivations Acting on Multilinear Polynomials in Prime Rings and Banach Algebras

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Abstract

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, F and G, the two nonzero 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

$\begin{aligned} F(G(f(x_1,\ldots ,x_n))f(x_1,\ldots ,x_n))=0 \end{aligned}$
for all $x_1,\ldots ,x_n \in I$, then one of the followings holds: (1) there exist $a,b\in U$ such that $F(x)=ax$ and $G(x)=bx$ for all $x\in R$ with $ab=0$; (2) there exist $a,b,p\in U$ such that $F(x)=ax+xb$ and $G(x)=px$ for all $x\in R$ with $F(p)=0$ and $f(x_1,\ldots ,x_n)^2$ is central valued on R. We also obtain some related results in cases where R is a semiprime ring and Banach algebra.

Keywords

Prime ring / Derivation / Generalized derivation / Extended centroid / Utumi quotient ring / Banach algebra

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Basudeb Dhara, Nurcan Argaç. Generalized Derivations Acting on Multilinear Polynomials in Prime Rings and Banach Algebras. Communications in Mathematics and Statistics, 2016, 4(1): 39-54 DOI:10.1007/s40304-015-0073-y

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