A note on generalized derivations as a Jordan homomorphisms

Bull. Korean Math. Soc. 2020 Vol. 57, No. 3, 709-737 https://doi.org/10.4134/BKMS.b190429 Published online February 13, 2020 Printed May 31, 2020

Arusha Chandrasekhar, Shailesh Kumar Tiwari Indian Institute of Science Education and Research, Bhopal; Indian Institute of Technology Patna

Abstract : Let $R$ be a prime ring of characteristic different from $2$. Suppose that $F$, $G$, $H$ and $T$ are generalized derivations of $R$. Let $U$ be the Utumi quotient ring of $R$ and $C$ be the center of $U$, called the extended centroid of $R$ and let $f(x_1,\ldots,x_n)$ be a non central multilinear polynomial over $C$. If \begin{align*} &\ F(f(r_1,\ldots,r_n))G(f(r_1,\ldots,r_n))-f(r_1,\ldots,r_n)T(f(r_1,\ldots,r_n))\\ =&\ H(f(r_1,\ldots,r_n)^2) \end{align*} for all $r_1, \ldots, r_n \in R$, then we describe all possible forms of $F$, $G$, $H$ and $T$.